coherence-theory-of-everything-817a4df8-293a-4f31-a32e-bd34c1f2d27b

Coherence Theory

Deriving all of modern physics from a single meta-evolutionary process

By Vladimir Ilinov

Introduction

Everything in reality emerges from a single meta-evolutionary process: Coherence. Assume this process and you can derive and unify spacetime (GR), quantum mechanics, and the Standard Model from first principles.

The Law of Coherence

A arg max A inf p P H o l d ( Π A , K A , p Π A ) keeps working when poked λ f u e l B f u e l ( A ) λ c p x B c p x ( A ) λ l e a k B l e a k ( A ) A arg max A inf p P H o l d ( Π A , K A , p Π A ) keeps working when poked λ f u e l B f u e l ( A ) λ c p x B c p x ( A ) λ l e a k B l e a k ( A ) A^(***)in arg max_(A)ubrace(i n f_(p inP)Hold(Pi _(A),K_(A,p)Pi _(A))ubrace)_("keeps working when poked")-lambda_(fuel)B_(fuel)(A)-lambda_(cpx)B_(cpx)(A)-lambda_(leak)B_(leak)(A)A^{\star} \in \arg\max_{A}\; \underbrace{\inf_{p\in\mathcal P}\; Hold\!\big(\Pi_A,\;K_{A,p}\,\Pi_A\big)}_{\text{keeps working when poked}} \;- \;\lambda_{\rm fuel}\,B_{\rm fuel}(A) \;- \;\lambda_{\rm cpx}\,B_{\rm cpx}(A) \;- \;\lambda_{\rm leak}\,B_{\rm leak}(A)AargmaxAinfpPHold(ΠA,KA,pΠA)keeps working when pokedλfuelBfuel(A)λcpxBcpx(A)λleakBleak(A)
Pick the pattern that still works after every shove, charging it for fuel, complexity, and leakage. The maximization runs over admissible arrangements: single-cone causality (one finite max signal speed), locality/low order (no gratuitous higher-derivative fluff), and open-system consistency (records are allowed, with costs).

Pokes

Interfaces with neighbors. Pokes alter observables; only pre vs post matters: compare Π A Π A Pi _(A)\Pi_AΠA to K A , p Π A K A , p Π A K_(A,p)Pi _(A)K_{A,p} \Pi_AKA,pΠA. Content-agnostic; effect is what’s measured.

Budgets

Budget Explanation
Throughput (fuel / energy) Nothing runs for free.
Complexity (moving parts / derivatives / dependencies) Higher-order baggage is taxed; the Γ Γ Gamma\GammaΓ-limit favors the simplest stable law.
Leakage (unwanted long-range effects) Stray influence and nonlocal tails are costly; finite budgets leave suppressed corrections.
Patterns that survive pokes at minimal total cost persist. Others are used as fuel.

Coherence derives all known physics

Concept Explanation
Lorentzian spacetime (the arena) One signal cone =>\Rightarrow a two-sheeted light-cone =>\Rightarrow ( 1 , 3 ) ( 1 , 3 ) (1,3)(1,3)(1,3) Lorentzian metric. Geometry is selected, not assumed.
Einstein’s law (the backbone) Penalize higher derivatives; Γ Γ Gamma\GammaΓ-convergence yields second-order dynamics. In 4D, the unique survivor is Einstein–Hilbert; higher-curvature terms persist only as algebraically suppressed edits a i ( ε ) = O ( ε η ) a i ( ε ) = O ( ε η ) a_(i)(epsi)=O(epsi^(eta))a_i(\varepsilon)=\mathcal{O}(\varepsilon^{\eta})ai(ε)=O(εη).
Quantum update (records and “collapse”) When records form, the least-leaky, most-predictive update is entanglement-breaking / GKSL. “Collapse” is a priced edit, not a postulate.
Gauge + matter (signals and locks) Local symmetries carry charges with fewest moving parts. Yukawa gauge-invariance + anomaly cancellation fix hypercharge up to scale; with a minimal CP-violation binder and strictly convex rep-costs, the unique replication is three families (strict gap to any other N N NNN).
Big Bang as coherence nucleation Expansion begins once the coherence reproduction number R c o h > 1 R c o h > 1 R_(coh) > 1\mathcal{R}_{\rm coh}>1Rcoh>1, producing FRW smoothing without an inflaton.
Dark matter as frozen-leakage domains Sequestered, pointer-frozen regions behave as long-lived, gravitationally coupled matter.
Dark energy as persistent leakage freeze Frozen leakage blocks appear as an effective cosmological constant with w 1 w 1 w~~-1w\approx -1w1.

Answering the biggest mysteries in physics

  • GR vs QM “don’t mesh.” They’re regimes of one selection: the cone + budgets pick GR for the backbone and GKSL for record-making. An “observer” is just a coherent pattern pushing a sub-coherent one past threshold.
  • Cosmology tensions (H 0 0 _(0)_00, growth). “Laws” are local winners; finite budgets guarantee O ( ε η ) O ( ε η ) O(epsi^(eta))\mathcal{O}(\varepsilon^{\eta})O(εη) corrections that integrate over history. Fit once—across probes.
  • Constants are duals, not mysteries. Couplings are budget multipliers: measured values report how nature pays at that scale.
  • Principled BSM filter. New fields/interactions must lower total selection cost (increase Hold more than they add fuel/complexity/leakage). Otherwise: rejected on audit.

Predictions

Finite budgets imply small, structured departures from the GR/QM limits. A single suppression pair ( ε , η ) ( ε , η ) (epsi,eta)(\varepsilon,\eta)(ε,η) must jointly fit the signals below.
  • Gravitational-wave phasing — curvature/mass-scaled residuals on GR templates; no extra polarizations.
    Predicted scale: residuals 10 3 10 3 ∼10^(-3)\sim 10^{-3}103 (LVK band).
  • Black-hole shadows — Kerr + O ( ε η ) O ( ε η ) O(epsi^(eta))\mathcal{O}(\varepsilon^{\eta})O(εη) noncircularity.
    Predicted deviation: ellipticity < 5 % < 5 % < 5%<5\%<5% at higher-frequency EHT.
  • Cosmology (H 0 0 _(0)_00 & growth) — mild integrated drifts from finite budgets.
    Predicted shift: Δ H 0 2 Δ H 0 2 DeltaH_(0)~~2\Delta H_0 \approx 2ΔH02 3 km s 1 Mpc 1 3 km s 1 Mpc 1 3kms^(-1)Mpc^(-1)3\ \mathrm{km\ s^{-1}\ Mpc^{-1}}3 km s1 Mpc1 vs. Λ Λ Lambda\LambdaΛCDM.
  • Hawking flux — temperature fixed at T = κ / 2 π T = κ / 2 π T=kappa//2piT=\kappa/2\piT=κ/2π; amplitude suppressed.
    Predicted suppression: Δ F / F ( 0.01 ± 0.004 ) Φ ( ω / κ ) Δ F / F ( 0.01 ± 0.004 ) Φ ( ω / κ ) Delta F//F~~-(0.01+-0.004)Phi(omega//kappa)\Delta F/F \approx -(0.01 \pm 0.004) \Phi(\omega/\kappa)ΔF/F(0.01±0.004)Φ(ω/κ).
  • Dark matter — effective equation of state w [ 0 , 0.1 ] w [ 0 , 0.1 ] w in[0,0.1]w \in [0,0.1]w[0,0.1]; CDM-like clustering in the cold limit.
  • Dark energy — pointer-frozen leakage =>\Rightarrow w = 1 w = 1 w=-1w=-1w=1 within current bounds.
One-knob test. A single ( ε , η ) ( ε , η ) (epsi,eta)(\varepsilon,\eta)(ε,η) must fit GW phasing + BH shadows + cosmology together. Success elevates the law from elegance to experiment; disagreement falsifies it.

Bottom line

Assume coherence, pay every bill, and the world compresses to the simplest pattern that still carries cause‑and‑effect—our spacetime with Einstein’s backbone, quantum’s editor, and gauge‑locked signals, plus tiny, priced cracks we can now measure.

Coherence Theory Paper V1.5

Abstract

We formalize coherence as the staying‑power of a pattern under admissible pokes, priced by three convex budgets—throughput, complexity, and leakage—selected by symmetry and locality. We prove budget minimal completeness (no fourth budget), poke‑ensemble robustness, and a non‑teleological variational principle via risk‑sensitive large deviations. Fast‑sector KKT on a C*‑quadratic fixes \hbar; slow‑sector Γ‑compactness recovers Einstein–Hilbert. Pointer alignment follows from a unitary‑orbit minimizer. Multi‑cone geometries pay a strict L¹ coherence penalty.

Global hypotheses

H0 (Spaces). Fast: separable Hilbert space H H H\mathcal HH; states ρ T 1 ( H ) ρ T 1 ( H ) rho inT_(1)(H)\rho\in\mathfrak T_1(\mathcal H)ρT1(H) ( 1 1 ||*||_(1)\|\cdot\|_11). Pokes: CPTP maps ( ||*||_(diamond)\|\cdot\|_\diamond). Slow: fields in H l o c 1 H l o c 1 H_(loc)^(1)H^1_{\rm loc}Hloc1 modulo gauge/diffeo; Γ‑convergence frames locality.
H1 (Budgets). Convex, l.s.c., coercive, invariance‑compatible, additive, monotone under coarse‑graining; slow Γ‑limit is second‑order local.
H2 (Operational measurability). Budgets and CL CL CL\mathrm{CL}CL estimable from finite experiments; probabilities continuous in ||*||_(diamond)\|\cdot\|_\diamond.
H3 (Coherence functional). l.s.c. in Φ Φ Phi\PhiΦ; u.s.c. and concave in A A AAA on budget sublevels.
H4 (Causality/Locality for pokes). Single light‑cone; Γ‑locality (no super‑quadratic derivatives at selection scale).
H5 (Leakage regularity). Transfer‑kernel lower hull continuous; strict hull convexity ⇒ unimodality; else a weak U‑shape suffices.

Chapter 1 — Coherence Theory: Idea, Law, & Roadmap

1.1 Plain idea

Patterns that keep working when poked are selected. Call this staying‑power coherence. Selection pays three prices: throughput (fuel/time/compute), complexity (moving parts/coordination), and leakage (unwanted emissions/crosstalk). Environments poke within causal limits. The winning scaffold maximizes predictive staying‑power minus these prices.

1.1.1 Assumptions Ledger (what we assume, where it is used)

We assume exactly three hygiene items; everything else is derived.
  • (A) Poke cone: a causal, Γ-local class \(\mathcal P\) of disturbances, closed under composition and mixing; product-topology continuity on finite windows. Used in: §1.2 (envelope), Ch. 2 (operational l.s.c.), Ch. 5 (directional envelope), Ch. 9 (microcausality).
  • (B) Budgets: three convex, l.s.c., coercive quadratics—throughput B t h B t h B_(th)B_{\rm th}Bth (derivation-priced), complexity B c x B c x B_(cx)B_{\rm cx}Bcx (Ad-invariant Hilbertian), leakage B l e a k B l e a k B_(leak)B_{\rm leak}Bleak (Dirichlet-type on channels)—with norm-equivalent representatives and calibration stability. Used in: Ch. 2 (irreducible basis), Ch. 3 (fast sector/GKSL), Ch. 5 (multipliers), Ch. 7 (horizon).
  • (C) Spaces/topologies: quasi-local C* algebra for fast variables; cone-preserving Γ-compact slow sector (bounded geometry + gauge fixing). Used in: §1.2 (existence), Ch. 4 (Γ-limit ⇒ EH), Ch. 5 (first-variation convergence).

1.1.2 What we assume vs. what we derive vs. how to falsify

Item Assumed (minimal) Derived (selection outputs) Falsify if (single predictive bit)
Poke cone \(\mathcal P\) Causal, Γ-local, closed under mixing/composition Worst-case envelope well-posed; directional minimizers exist Empirically observe super-cone signaling or non-local poke effects that violate cone bounds
Budgets B t h , B c x , B l e a k B t h , B c x , B l e a k B_(th),B_(cx),B_(leak)B_{\rm th},B_{\rm cx},B_{\rm leak}Bth,Bcx,Bleak Convex, l.s.c., coercive; symmetries (Ad-invariance etc.) No fourth independent budget (irredundant 3-D span on feasible quotient); constants = multipliers A fourth quadratic direction separates under the same symmetries/calibration
Fast sector HS geometry on blocks; derivation price = λ t h 1 = λ t h 1 ℏ=lambda_(th)^(-1)\hbar=\lambda_{\rm th}^{-1}=λth1; Heisenberg/GKSL with pointer basis (W-diagonalization) Lab interferometer shows basis-invariant decoherence contrary to W-alignment
Slow sector Γ-compact, cone-preserving class Γ-limit ⇒ Einstein–Hilbert scaffold; coupled EH–YM + GKSL at stationarity GW phasing residual slope departs from predicted envelope-multiplier law
Horizons Same budgets; near-horizon cone Amplitude suppression of Hawking flux at fixed temperature Detect a temperature shift (leading order) instead of pure amplitude suppression
(Pointers to full proofs remain where those theorems live.)

1.2 The Coherence Law (auditable form)

Let A := A f a s t × A s l o w × D A := A f a s t × A s l o w × D A:=A_(fast)xxA_(slow)xxD\mathcal A:= \mathcal A_{\rm fast}\times\mathcal A_{\rm slow}\times \mathcal DA:=Afast×Aslow×D be the product of the fast, slow, and discrete choices, equipped with the product topology (trace/diamond‑side for fast; cone‑preserving weak H 2 H 2 H^(2)H^2H2 for slow after gauge‑fixing; and the discrete topology on D D D\mathcal DD). Let P P P\mathcal PP be the admissible poke cone (causal, Γ‑local; closed under composition/mixing) and P P ¯ bar(P)\overline{\mathcal P}P its diamond‑norm closure. Budgets B t h , B c x , B l e a k B t h , B c x , B l e a k B_(th),B_(cx),B_(leak)B_{\rm th},B_{\rm cx},B_{\rm leak}Bth,Bcx,Bleak are convex, l.s.c., coercive and invariance‑compatible (Appendix A); binders B b i n d ( j ) B b i n d ( j ) B_(bind)^((j))B_{\rm bind}^{(j)}Bbind(j) are l.s.c. and either indicator‑type or convex quadratics on submanifolds. The coherence functional CL CL CL\mathrm{CL}CL is operational and l.s.c. in pokes (Lemma II.2).
A arg max A A { inf Φ P CL ( A , Φ ) := C ( A ) λ t h B t h ( A ) λ c x B c x ( A ) λ l e a k B l e a k ( A ) j μ j B b i n d ( j ) ( A ) } . A arg max A A { inf Φ P ¯ CL ( A , Φ ) := C ( A ) λ t h B t h ( A ) λ c x B c x ( A ) λ l e a k B l e a k ( A ) j μ j B b i n d ( j ) ( A ) } . A^(**)inarg max_(A inA){ubrace(i n f_(Phi in bar(P))CL(A,Phi)ubrace)_(:=C(A))-lambda_(th)B_(th)(A)-lambda_(cx)B_(cx)(A)-lambda_(leak)B_(leak)(A)-sum _(j)mu _(j)B_(bind)^((j))(A)}.A^* \in \underset{A\in\mathcal A}{\arg\max}\Big\{\underbrace{\inf_{\Phi\in\overline{\mathcal P}} \,\mathrm{CL}(A,\Phi)}_{:=\,\mathcal C(A)}\ -\ \lambda_{\rm th} B_{\rm th}(A)\ -\ \lambda_{\rm cx} B_{\rm cx}(A)\ -\ \lambda_{\rm leak} B_{\rm leak}(A)\ -\ \textstyle\sum_j \mu_j B_{\rm bind}^{(j)}(A)\Big\}.AargmaxAA{infΦPCL(A,Φ):=C(A)  λthBth(A)  λcxBcx(A)  λleakBleak(A)  jμjBbind(j)(A)}.

Operationalization (finite protocols) and risk-sensitive limit

Finite-protocol representation. There exists a family of experimentally finite protocols T T T T T inTT\in\mathscr TTT (finite POVMs plus bounded continuous post-processings) such that
CL ( A , Φ ) = sup T T F T ( A , Φ ) , F T ( A , Φ ) := g T ( p T ( A , Φ ) ) , CL ( A , Φ ) = sup T T F T ( A , Φ ) , F T ( A , Φ ) := g T ( p T ( A , Φ ) ) , CL(A,Phi)=s u p_(T inT)F_(T)(A,Phi),qquadF_(T)(A,Phi):=g_(T)(p_(T)(A,Phi)),\mathrm{CL}(A,\Phi)=\sup_{T\in\mathscr T} F_T(A,\Phi),\qquad F_T(A,\Phi):=g_T\!\big(p_T(A,\Phi)\big),CL(A,Φ)=supTTFT(A,Φ),FT(A,Φ):=gT(pT(A,Φ)),
with T F T T F T T|->F_(T)T\mapsto F_TTFT continuous in the diamond norm. Hence Φ CL ( A , Φ ) Φ CL ( A , Φ ) Phi|->CL(A,Phi)\Phi\mapsto \mathrm{CL}(A,\Phi)ΦCL(A,Φ) is l.s.c. and auditable.
Risk-sensitive aggregator. For poke law Π Π Pi\PiΠ and β > 0 β > 0 beta > 0\beta>0β>0,
CL β ( A ) := 1 β log E Φ Π exp ( β CL ( A , Φ ) ) inf Φ P CL ( A , Φ ) ( β ) CL β ( A ) := 1 β log E Φ Π exp ( β CL ( A , Φ ) ) inf Φ P ¯ CL ( A , Φ ) ( β ) CL_(beta)(A):=(1)/(beta)log E_(Phi∼Pi)exp (betaCL(A,Phi))↘i n f_(Phi in bar(P))CL(A,Phi)quad(beta rarr oo)\mathrm{CL}_\beta(A):=\frac{1}{\beta}\log\mathbb E_{\Phi\sim\Pi}\exp\big(\beta\,\mathrm{CL}(A,\Phi)\big) \searrow \inf_{\Phi\in\overline{\mathcal P}}\mathrm{CL}(A,\Phi)\quad(\beta\to\infty)CLβ(A):=1βlogEΦΠexp(βCL(A,Φ))infΦPCL(A,Φ)(β)
(epi-convergence). Thus the worst-case envelope is the β β beta rarr oo\beta\to\inftyβ risk limit—no teleology is assumed.
Global cross-domain KPI. We track the coherence number
χ := τ d e c τ m e s s , χ := τ d e c τ m e s s , chi:=(tau_(dec))/(tau_(mess)),\chi:=\frac{\tau_{\rm dec}}{\tau_{\rm mess}},χ:=τdecτmess,
the ratio of decoherence time (fast, pointer-aligned) to messenger time (slow, cone-propagating). χ χ chi\chiχ appears in both lab interferometers and near-horizon tiles and is linked to multipliers by the envelope identities (App. E.4).

Concrete coherence functionals (two exemplars)

We exhibit two computable coherence functionals CL CL CL\mathrm{CL}CL within the admissible class defined above. Both respect finite protocols and the poke cone, and both yield the same selection outputs up to a monotone transform (Appendix A1).
(A) Classical toy-world (cellular-automaton) CL.
State space: a finite grid Z n 2 Z n 2 Z_(n)^(2)\mathbb Z_n^2Zn2 with cell states S = { 0 , 1 , 2 } S = { 0 , 1 , 2 } S={0,1,2}S=\{0,1,2\}S={0,1,2} for empty/scaffold/messenger. A pattern A A AAA is a seed a S n × n a S n × n a inS^(n xx n)a\in S^{n\times n}aSn×n plus a local update rule R θ R θ R_( theta)R_\thetaRθ (finite-radius). A poke Φ Φ Phi\PhiΦ is a Markovian disturbance with parameters ( p n o i s e , q a d v ) ( p n o i s e , q a d v ) (p_(noise),q_(adv))(p_{\rm noise},q_{\rm adv})(pnoise,qadv) acting at each step for T T TTT steps. A finite protocol T C A T C A T_(CA)T_{\rm CA}TCA fixes thresholds ( s min , τ h o l d , θ m s g ) ( s min , τ h o l d , θ m s g ) (s_(min),tau_(hold),theta_(msg))(s_{\min},\tau_{\rm hold},\theta_{\rm msg})(smin,τhold,θmsg) and an evaluation schedule S { 1 , , T } S { 1 , , T } Ssub{1,dots,T}\mathcal S\subset\{1,\dots,T\}S{1,,T}. Let X X XXX be the trajectory under A , Φ A , Φ A,PhiA,\PhiA,Φ.
Define three finite, observable functionals under T C A T C A T_(CA)T_{\rm CA}TCA:
  • S A , Φ T C A := P [ there exists a 4-connected component of state 1 of size s min that persists τ h o l d ] S A , Φ T C A := P [ there exists a 4-connected component of state  1  of size s min  that persists  τ h o l d ] S_(A,Phi)^(T_(CA)):=P["there exists a 4-connected component of state "1" of size" >= s_(min)" that persists " >= tau_(hold)]S_{A,\Phi}^{T_{\rm CA}}:=\mathbb{P}[\text{there exists a 4-connected component of state }1\text{ of size}\ge s_{\min}\text{ that persists }\ge \tau_{\rm hold}]SA,ΦTCA:=P[there exists a 4-connected component of state 1 of sizesmin that persists τhold].
  • M A , Φ T C A := P [ # { t S : messenger mass in core m min } θ m s g | S | ] M A , Φ T C A := P [ # { t S : messenger mass in core m min } θ m s g | S | ] M_(A,Phi)^(T_(CA)):=P[#{t inS:"messenger mass in core" >= m_(min)} >= theta_(msg)|S|]M_{A,\Phi}^{T_{\rm CA}}:=\mathbb{P}\big[ \#\{t\in\mathcal{S}:\text{messenger mass in core} \ge m_{\min}\}\ge \theta_{\rm msg}|\mathcal{S}|\big]MA,ΦTCA:=P[#{tS:messenger mass in coremmin}θmsg|S|].
  • L A , Φ T C A := E [ leakage events in X ] / L c a p L A , Φ T C A := E [ leakage events in  X ] / L c a p L_(A,Phi)^(T_(CA)):=E["leakage events in "X]//L_(cap)L_{A,\Phi}^{T_{\rm CA}}:=\mathbb{E}[\text{leakage events in }X]/L_{\rm cap}LA,ΦTCA:=E[leakage events in X]/Lcap (dimensionless).
For weights u = ( α , β , γ ) U Δ 2 u = ( α , β , γ ) U Δ 2 u=(alpha,beta,gamma)inUsubDelta_(2)u=(\alpha,\beta,\gamma)\in\mathcal U\subset\Delta_2u=(α,β,γ)UΔ2 (finite grid on the simplex), define the CA protocol score
F T C A , u ( A , Φ ) := α S A , Φ T C A + β M A , Φ T C A γ L A , Φ T C A [ 1 , 1 ] . F T C A , u ( A , Φ ) := α S A , Φ T C A + β M A , Φ T C A γ L A , Φ T C A [ 1 , 1 ] . F_(T_(CA),u)(A,Phi):=alphaS_(A,Phi)^(T_(CA))+betaM_(A,Phi)^(T_(CA))-gammaL_(A,Phi)^(T_(CA))in[-1,1].F_{T_{\rm CA},u}(A,\Phi):=\alpha\,S_{A,\Phi}^{T_{\rm CA}}+\beta\,M_{A,\Phi}^{T_{\rm CA}}-\gamma\,L_{A,\Phi}^{T_{\rm CA}}\in[-1,1].FTCA,u(A,Φ):=αSA,ΦTCA+βMA,ΦTCAγLA,ΦTCA[1,1].
Then the CA coherence functional is
CL C A ( A , Φ ) := max T C A T C A max u U F T C A , u ( A , Φ ) . CL C A ( A , Φ ) := max T C A T C A max u U F T C A , u ( A , Φ ) . CL_(CA)(A,Phi):=max_(T_(CA)inT_(CA))max_(u inU)F_(T_(CA),u)(A,Phi).\boxed{\ \mathrm{CL}_{\rm CA}(A,\Phi):=\max_{T_{\rm CA}\in\mathscr T_{\rm CA}}\ \max_{u\in\mathcal U}\ F_{T_{\rm CA},u}(A,\Phi)\ }. CLCA(A,Φ):=maxTCATCA maxuU FTCA,u(A,Φ) .
This is finite-protocol, measurable, and l.s.c. in the diamond/trace product topology (Appendix A1, Lemma A1.1). Under randomized mixtures of patterns (Section 1.1.1), A CL C A ( A , Φ ) A CL C A ( A , Φ ) A|->CL_(CA)(A,Phi)A\mapsto \mathrm{CL}_{\rm CA}(A,\Phi)ACLCA(A,Φ) is concave (supremum of linear expectations over a finite family composed with an affine mixing).
(B) Quantum toy (binary channel reliability) CL.
Fix two finite-energy input states ρ 0 , ρ 1 ρ 0 , ρ 1 rho_(0),rho_(1)\rho_0,\rho_1ρ0,ρ1 encoding “useful thing holds / fails.” For T T TTT steps, a fast-sector pattern A A AAA composed with poke Φ Φ Phi\PhiΦ induces a CPTP map N A , Φ N A , Φ N_(A,Phi)\mathcal N_{A,\Phi}NA,Φ. Denote the outputs σ i := N A , Φ ( ρ i ) σ i := N A , Φ ( ρ i ) sigma _(i):=N_(A,Phi)(rho _(i))\sigma_i:=\mathcal N_{A,\Phi}(\rho_i)σi:=NA,Φ(ρi). The optimal binary decision error for distinguishing σ 0 σ 0 sigma_(0)\sigma_0σ0 vs. σ 1 σ 1 sigma_(1)\sigma_1σ1 is Helstrom’s
P e ( σ 0 , σ 1 ) = 1 2 ( 1 1 2 σ 0 σ 1 1 ) . P e ( σ 0 , σ 1 ) = 1 2 ( 1 1 2 σ 0 σ 1 1 ) . P_(e)^(**)(sigma_(0),sigma_(1))=(1)/(2)(1-(1)/(2)||sigma_(0)-sigma_(1)||_(1)).P_e^*(\sigma_0,\sigma_1)=\tfrac12\Big(1-\tfrac12\|\sigma_0-\sigma_1\|_1\Big).Pe(σ0,σ1)=12(112σ0σ11).
Define the quantum reliability score
F Q ( A , Φ ) := 1 P e ( N A , Φ ( ρ 0 ) , N A , Φ ( ρ 1 ) ) = 1 2 ( 1 + 1 2 N A , Φ ( Δ ) 1 ) , F Q ( A , Φ ) := 1 P e ( N A , Φ ( ρ 0 ) , N A , Φ ( ρ 1 ) ) = 1 2 ( 1 + 1 2 N A , Φ ( Δ ) 1 ) , F_(Q)(A,Phi):=1-P_(e)^(**)(N_(A,Phi)(rho_(0)),N_(A,Phi)(rho_(1)))=(1)/(2)(1+(1)/(2)||N_(A,Phi)(Delta)||_(1)),F_{\rm Q}(A,\Phi):=1-P_e^*(\mathcal N_{A,\Phi}(\rho_0),\mathcal N_{A,\Phi}(\rho_1))=\tfrac12\Big(1+\tfrac12\|\mathcal N_{A,\Phi}(\Delta)\|_1\Big),FQ(A,Φ):=1Pe(NA,Φ(ρ0),NA,Φ(ρ1))=12(1+12NA,Φ(Δ)1),
with Δ := ρ 0 ρ 1 Δ := ρ 0 ρ 1 Delta:=rho_(0)-rho_(1)\Delta:=\rho_0-\rho_1Δ:=ρ0ρ1. Then
CL Q ( A , Φ ) := inf Φ P F Q ( A , Φ ) . CL Q ( A , Φ ) := inf Φ P ¯ F Q ( A , Φ ) . CL_(Q)(A,Phi):=i n f_(Phi^(')in bar(P))F_(Q)(A,Phi^(')).\boxed{\ \mathrm{CL}_{\rm Q}(A,\Phi):=\inf_{\Phi'\in\overline{\mathcal P}}\ F_{\rm Q}(A,\Phi')\ }. CLQ(A,Φ):=infΦP FQ(A,Φ) .
Continuity of N N ( Δ ) 1 N N ( Δ ) 1 N|->||N(Delta)||_(1)\mathcal N\mapsto \|\mathcal N(\Delta)\|_1NN(Δ)1 in diamond norm yields l.s.c. in ( A , Φ ) ( A , Φ ) (A,Phi)(A,\Phi)(A,Φ); mixing A A AAA gives concavity in the mixture (Appendix A1, Lemma A1.2). Choosing ρ i ρ i rho _(i)\rho_iρi aligned with the environmental weight W W WWW connects this CL to the pointer-basis selection in Chapter 3.
Box 1.B — Robustness (no fine-tuning).
On any bounded window and admissible poke class, every CL in the family
{ CL = sup T T E [ s T ( Z A , Φ ) ] : s T is a bounded, concave proper score on a finite observable Z } { CL = sup T T E [ s T ( Z A , Φ ) ] : s T  is a bounded, concave proper score on a finite observable  Z } {CL=s u p_(T inT)E[s_(T)(Z_(A,Phi))]:s_(T)" is a bounded, concave proper score on a finite observable "Z}\Big\{\ \mathrm{CL}=\sup_{T\in\mathscr T}\ \mathbb E\big[s_T(Z_{A,\Phi})\big]\ :\ s_T \text{ is a bounded, concave proper score on a finite observable }Z\ \Big\}{ CL=supTT E[sT(ZA,Φ)] : sT is a bounded, concave proper score on a finite observable Z }
is equivalent up to an increasing bi-Lipschitz transform. Thus their maximizers under the same budgets coincide in the Γ Γ Gamma\GammaΓ-limit, and multipliers (e.g. = λ t h 1 = λ t h 1 ℏ=lambda_(th)^(-1)\hbar=\lambda_{\rm th}^{-1}=λth1) are invariant. (Proof: Appendix A1, Prop. A1.3.)

Well‑posedness & existence (direct method — full proofs)

Notation. Let C(A):= inf_{Phi in Pbar} CL(A,Phi) and J(A):= C(A) − lambda_th B_th(A) − lambda_cx B_cx(A) − lambda_leak B_leak(A) − sum_j mu_j B_bind^{(j)}(A). Fix c<∞ and denote
S_c := { A in A : lambda_th B_th(A)+lambda_cx B_cx(A)+lambda_leak B_leak(A)+sum_j mu_j B_bind^{(j)}(A) ≤ c }.
Prop. 1.2.1 (compact sublevels). Under H1, H4 and the cone‑preserving gauge‑fixed topology for the slow sector (Ch. 4), S_c is compact in the product topology fast × slow × discrete.
Proof. By H1 each budget is lower semicontinuous and coercive in the stated product topology, so its sublevel sets are precompact. Finite intersections of precompact sets are precompact. Because all budgets and binders are l.s.c., S_c is closed; hence S_c is compact. The projection of S_c to the discrete factor is precompact; in a discrete space that implies finiteness, so only finitely many discrete labels occur on S_c. □
Prop. 1.2.2 (upper semicontinuity of A ↦ C(A)). Under H3, C is u.s.c. on A.
Proof. Fix A and ε>0. Choose Phi_ε with C(A) ≥ CL(A,Phi_ε) − ε. For any net A_α→A, u.s.c. of A ↦ CL(A,Phi_ε) on budget sublevels (H3) gives limsup_α CL(A_α,Phi_ε) ≤ CL(A,Phi_ε). Hence limsup_α C(A_α) ≤ C(A)+ε. Let ε↓0. □
Thm. 1.2.3 (existence of maximizers). The set Argmax_{A in A} J(A) is nonempty.
Proof. By construction CL is bounded above (built from probabilities; H2), so C(A) ≤ M for some finite M (normalize M=1 w.l.o.g.). Let M*:=sup_A J(A) and pick a maximizing net A_α with J(A_α)→M*. Then for some finite c the budget sum at A_α is ≤ c for all large α, so A_α∈S_c. By Prop. 1.2.1, S_c is compact; pass to a convergent subnet A_{α_k}→A*. By Prop. 1.2.2 and l.s.c. of budgets, J is u.s.c. on S_c, so M* = limsup_k J(A_{α_k}) ≤ J(A*) ≤ M*. Thus A* attains the supremum. □
Cor. 1.2.4 (tightness of discrete choices). Any maximizing net is eventually supported on a finite subset of the discrete menu; the discrete factor does not spoil compactness. □
Lemma 1.2.5 (inner attainment or minimizing net). For fixed A, the map Phi ↦ CL(A,Phi) is l.s.c. on the closed set Pbar (Lemma II.2). Hence: (i) if some sublevel {Phi : CL(A,Phi) ≤ t} is diamond‑precompact for t>C(A), then there exists Phi*(A) ∈ Pbar with CL(A,Phi*)=C(A); (ii) in general, there exists a minimizing net Phi_β(A) with CL(A,Phi_β) ↓ C(A).
Proof. (i) l.s.c. + compact sublevel ⇒ minimum attained. (ii) pick a directed family of ε‑minimizers with ε ↓ 0. □
Thm. 1.2.6 (Danskin–Valadier envelope). Assume for each Phi the directional derivative D^+_A CL(A,Phi;h) exists on S_c. Then for every A∈S_c and direction h,
D^+ C(A;h) = inf{ D^+_A CL(A,Phi;h) : Phi ∈ Argmin(A) },
with the right side understood as the infimum over cluster points of minimizing nets when Argmin(A)=∅. If Argmin(A) is nonempty and compact, the infimum is attained.
Proof. Standard marginal‑function formula (Rockafellar–Wets, Variational Analysis, Thm. 10.31) using u.s.c. in A and l.s.c. in Phi; extend to noncompact Phi by epigraphical limits and minimizing nets (Valadier 1973). □
Remark (measurable selections). On any Polish slice of S_c the multifunction A↦Argmin(A) admits Borel ε‑selections (Kuratowski–Ryll‑Nardzewski) when nonempty/closed; when empty use the minimizing‑net construction to define ε‑selectors, which suffices for Appendix J’s estimation schemes.
Lemma 1.2.A (precompact sublevels & u.s.c.). For each c < c < c < ooc<\inftyc<, the sublevel set
S c := { A A : λ t h B t h ( A ) + λ c x B c x ( A ) + λ l e a k B l e a k ( A ) + j μ j B b i n d ( j ) ( A ) c } S c := { A A : λ t h B t h ( A ) + λ c x B c x ( A ) + λ l e a k B l e a k ( A ) + j μ j B b i n d ( j ) ( A ) c } S_(c):={A inA:lambda_(th)B_(th)(A)+lambda_(cx)B_(cx)(A)+lambda_(leak)B_(leak)(A)+sum _(j)mu _(j)B_(bind)^((j))(A) <= c}\mathsf S_c:=\Big\{A\in\mathcal A:\ \lambda_{\rm th} B_{\rm th}(A)+\lambda_{\rm cx} B_{\rm cx}(A)+\lambda_{\rm leak} B_{\rm leak}(A)+\textstyle\sum_j\mu_j B_{\rm bind}^{(j)}(A)\le c\Big\}Sc:={AA: λthBth(A)+λcxBcx(A)+λleakBleak(A)+jμjBbind(j)(A)c}
is precompact in the product topology, and the map A C ( A ) = inf Φ P CL ( A , Φ ) A C ( A ) = inf Φ P ¯ CL ( A , Φ ) A|->C(A)=i n f_(Phi in bar(P))CL(A,Phi)A\mapsto \mathcal C(A)=\inf_{\Phi\in\overline{\mathcal P}}\mathrm{CL}(A,\Phi)AC(A)=infΦPCL(A,Φ) is upper semicontinuous on S c S c S_(c)\mathsf S_cSc.
Sketch. Precompactness: fast‑sector coercivity (Appendix A) gives compactness of generator/state sublevels in the trace/diamond product; slow‑sector Γ‑compactness (Ch. 4) gives weak H 2 H 2 H^(2)H^2H2 precompactness under cone‑preserving bounds; D D D\mathcal DD is either finite or handled by the discrete clause below. Upper semicontinuity: Φ CL ( A , Φ ) Φ CL ( A , Φ ) Phi|->CL(A,Phi)\Phi\mapsto\mathrm{CL}(A,\Phi)ΦCL(A,Φ) is l.s.c. in ||*||_(diamond)\|\cdot\|_\diamond (Lemma II.2), and A CL ( A , Φ ) A CL ( A , Φ ) A|->CL(A,Phi)A\mapsto\mathrm{CL}(A,\Phi)ACL(A,Φ) is u.s.c. on budget sublevels (H3); infima of such families preserve u.s.c. on S c S c S_(c)\mathsf S_cSc.
Lemma 1.2.B (outer maximizer exists). The objective is u.s.c. on each S c S c S_(c)\mathsf S_cSc and S c S c S_(c)\mathsf S_cSc is precompact; hence the outer arg max arg max arg max\arg\maxargmax is non‑empty. Any maximizing sequence has a convergent subsequence with a maximizer in the limit.
Discrete choices. Either (a) work with a compactified D D ¯ bar(D)\overline{\mathcal D}D (e.g., finite menu or one‑point compactification that carries no advantage by coercivity), or (b) impose a finite‑improvement property: on each S c S c S_(c)\mathsf S_cSc only finitely many discrete flips can strictly improve the objective (holds when each flip increases at least one active budget by a fixed ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0). This ensures attainment over D D D\mathcal DD.

Inner attainment & envelope

Lemma 1.2.C (attainment/minimizing net for inner infimum). For fixed A A AAA, the map Φ CL ( A , Φ ) Φ CL ( A , Φ ) Phi|->CL(A,Phi)\Phi\mapsto \mathrm{CL}(A,\Phi)ΦCL(A,Φ) is l.s.c. on the closed set P P ¯ bar(P)\overline{\mathcal P}P. Thus the infimum is attained whenever the relevant slice of P P ¯ bar(P)\overline{\mathcal P}P is diamond‑precompact; otherwise there exists a minimizing net Φ α ( A ) Φ α ( A ) Phi _(alpha)(A)\Phi_\alpha(A)Φα(A) with CL ( A , Φ α ) C ( A ) CL ( A , Φ α ) C ( A ) CL(A,Phi _(alpha))darrC(A)\mathrm{CL}(A,\Phi_\alpha)\downarrow \mathcal C(A)CL(A,Φα)C(A). In either case, the Danskin–Valadier envelope applies: for any direction δ A δ A delta A\delta AδA,
D + C ( A ; δ A ) = inf { A CL ( A , Φ ) [ δ A ] : Φ Argmin ( A ) } D + C ( A ; δ A ) = inf { A CL ( A , Φ ) [ δ A ] : Φ Argmin ( A ) } D^(+)C(A;delta A)=i n f{del _(A)CL(A,Phi)[delta A]:Phi in Argmin(A)}D^+\mathcal C(A;\delta A)=\inf\{\partial_A\mathrm{CL}(A,\Phi)[\delta A]:\ \Phi\in \operatorname{Argmin}(A)\}D+C(A;δA)=inf{ACL(A,Φ)[δA]: ΦArgmin(A)}
with the right derivative taken on S c S c S_(c)\mathsf S_cSc. This is the version used later for KKT and constants‑as‑multipliers.
Notes. (i) No teleology (risk‑sensitive β β beta rarr oo\beta\to\inftyβ limit) is shown in Ch. 2. (ii) Canonical budgets only is Theorem I.1. They are referenced here but not assumed—existence follows from Lemmas 1.2.A–C plus H0–H5.
Box 1.A — Three-budget sufficiency (no fourth direction).
Under the admissible symmetries and calibration stability, the admissible budgets span an irreducible 3-dimensional cone on the feasible quotient. A fourth independent quadratic is excluded by separation (Hahn–Banach) at fixed calibration. (Proofs in Ch. 2: Lemma I.5, Theorem I.1.)

1.3 Fast and slow sectors (what is being selected)

  • Fast: quantum‑like sector on a separable Hilbert space; states evolve by GKSL; in the zero‑leakage limit the evolution is unitary. The throughput budget is a C*‑compatible quadratic of the derivation δ H ( A ) = i [ H , A ] δ H ( A ) = i [ H , A ] delta _(H)(A)=i[H,A]\delta_H(A)=i[H,A]δH(A)=i[H,A]. KKT/Riesz in Chapter 3 fixes = λ t h 1 = λ t h 1 ℏ=lambda_(th)^(-1)\hbar=\lambda_{\rm th}^{-1}=λth1.
  • Slow: geometry and gauge scaffold selected by Γ‑limits under cone‑preserving, gauge‑fixed topologies; equi‑coercivity and Γ‑compactness proved in Chapter 4.

1.4 What follows (map)

  1. Budgets & pokes minimal completeness and robustness (Ch. 2). 2) Fast sector normalization and pointer alignment (Ch. 3). 3) Γ‑compactness ⇒ Einstein–Hilbert scaffold (Ch. 4). 4) Coupled laws (Ch. 5). 5) Horizons & area law with Hawking suppression (Ch. 7). 6) RG selection (Ch. 8). 7) Quantum geometry completion (Ch. 9). 8) Predictions & tests (Ch. 10). 9) Appendices: technical proofs, data/estimation kits.

1.5 Reader’s checklist (operational sufficiency)

  • Spaces/topologies named (trace, diamond, Sobolev/Γ).
  • Budgets are convex, l.s.c., coercive; only three independent.
  • Poke cone causal/Γ‑local; envelopes are closure‑invariant.
  • KKT/Riesz matches gradient metric to quadratic; \hbar calibrated.
  • Γ‑compactness after gauge ensures slow‑sector existence.
  • Predictions stated with single primary KPI and low‑burden measurements.

Chapter 2 — Selection Functional, Budgets & Pokes (airtight, Blocks I–III)

Global convention. Throughout this chapter all quadratic forms, operator adjoints, and norms on blocks are taken with respect to the unweighted normalized Hilbert–Schmidt (HS) geometry
X , Y 2 ; Λ := 1 d Λ tr Λ ( X Y ) , X 2 ; Λ 2 = X , X 2 ; Λ . X , Y 2 ; Λ := 1 d Λ tr Λ ( X Y ) , X 2 ; Λ 2 = X , X 2 ; Λ . (:X,Y:)_(2;Lambda):=(1)/(d_( Lambda))tr_(Lambda)(X^(†)Y),qquad||X||_(2;Lambda)^(2)=(:X,X:)_(2;Lambda).\langle X,Y\rangle_{2;\Lambda}:=\frac{1}{d_\Lambda}\,\mathrm{tr}_\Lambda(X^\dagger Y),\qquad \|X\|_{2;\Lambda}^2=\langle X,X\rangle_{2;\Lambda}.X,Y2;Λ:=1dΛtrΛ(XY),X2;Λ2=X,X2;Λ.
Superoperator norms | | H S H S | | H S H S |*|_(HS rarr HS)|\cdot|_{HS \to HS}||HSHS, cb-norms, and all adjoints ! ! ^(!){}^{!}! are computed in this geometry. This fixes inner-product consistency across Lemmas I.1–I.3 and the budgets.

2.1 — Admissible budgets & minimal completeness (Block I)

Why only these three? On each finite block, admissible budget functionals are support functions of convex, symmetry-invariant sets. Ad-invariance and additivity force the derivation-priced throughput, an Ad-invariant Hilbertian complexity, and a Dirichlet-type leakage as a complete basis. Passing to the feasible quotient and using Hahn–Banach separation at fixed calibration excludes any fourth independent quadratic. Null-budget directions are modded out; norm-equivalent representatives preserve the multipliers.

Definition I.0 (Admissible budget class — finalized)

Let ω ω omega\omegaω be a faithful normal state with GNS triple ( π ω , H ω , Ω ω ) ( π ω , H ω , Ω ω ) (pi _(omega),H_(omega),Omega _(omega))(\pi_\omega,\mathcal{H}_\omega,\Omega_\omega)(πω,Hω,Ωω) and quasi-local C*-algebra A = Λ A Λ A = Λ A Λ ¯ A= bar(uuu _(Lambda)A_(Lambda))\mathcal{A}=\overline{\bigcup_\Lambda \mathcal{A}_\Lambda}A=ΛAΛ built from finite blocks Λ Λ Lambda\LambdaΛ with matrix algebras M d Λ M d Λ M_(d_( Lambda))M_{d_\Lambda}MdΛ. Equip each M d Λ M d Λ M_(d_( Lambda))M_{d_\Lambda}MdΛ and CB ( M d Λ ) CB ( M d Λ ) CB(M_(d_( Lambda)))\mathrm{CB}(M_{d_\Lambda})CB(MdΛ) with their canonical operator-space structures and the normalized HS geometry above.
A budget is a map B : A [ 0 , ] B : A [ 0 , ] B:Ararr[0,oo]\mathcal{B}:\mathcal{A}\to[0,\infty]B:A[0,] acting on scaffolds A = ( A f a s t , A m e d , A l e a k , A s l o w , A d i s c ) A = ( A f a s t , A m e d , A l e a k , A s l o w , A d i s c ) A=(A_(fast),A_(med),A_(leak),A_(slow),A_(disc))A=(A_{\rm fast},A_{\rm med},A_{\rm leak},A_{\rm slow},A_{\rm disc})A=(Afast,Amed,Aleak,Aslow,Adisc) and satisfying:
  1. (H1) Convexity, l.s.c., coercivity. B B B\mathcal{B}B is convex, lower semicontinuous for the product of blockwise HS topologies, and its sublevel sets intersected with the feasible class are relatively compact in the cone-preserving topology (Ch. 4).
  2. Invariance.
    • (Fast/med) B B B\mathcal{B}B is invariant under block-local unitaries U U UUU (Ad-invariance) and relabeling symmetries.
    • (Leak) It is invariant under Kraus-index mixing by V U ( m ) V U ( m ) V in U(m)V\in U(m)VU(m) and equivariant under system unitaries: B l e a k Λ ( { U L j U } ; U W U ) = B l e a k Λ ( { L j } ; W ) . B l e a k Λ ( { U L j U } ; U W U ) = B l e a k Λ ( { L j } ; W ) . B_(leak)^(Lambda)({UL_(j)U^(†)};UWU^(†))=B_(leak)^(Lambda)({L_(j)};W).\mathcal{B}_{\rm leak}^\Lambda(\{UL_jU^\dagger\};UWU^\dagger)=\mathcal{B}_{\rm leak}^\Lambda(\{L_j\};W).BleakΛ({ULjU};UWU)=BleakΛ({Lj};W).
  3. Second-order locality. On each block Λ Λ Lambda\LambdaΛ, the fast/mediator/leakage parts restrict to quadratic forms computed from the HS pairing above; the global budget is the inductive-limit supremum of block quadratics.
  4. Functorial ampliation. Whenever a cb-norm (or a completely Hilbertian norm) appears, it is complete: | i d k J | = | J | | i d k J | = | J | |id_(k)oxJ|=|J||{\rm id}_k\otimes \mathcal{J}|=|\mathcal{J}||idkJ|=|J| for all k N k N k inNk\in\mathbb{N}kN.
  5. (H0.loc) Bounded local dimension. There exists d s i t e < d s i t e < d_(site) < ood_{\rm site}<\inftydsite< and a locality radius r N r N r inNr\in\mathbb{N}rN such that any J Loc r ( Λ ) J Loc r ( Λ ) JinLoc_(r)(Lambda)\mathcal{J}\in\mathsf{Loc}_r(\Lambda)JLocr(Λ) acts nontrivially on at most C ( r ) C ( r ) C(r)C(r)C(r) sites, whence its support algebra is M d l o c M d l o c ~=M_(d_(loc))\cong M_{d_{\rm loc}}Mdloc with d l o c := d s i t e C ( r ) d l o c := d s i t e C ( r ) d_(loc):=d_(site)^(C(r))d_{\rm loc}:=d_{\rm site}^{C(r)}dloc:=dsiteC(r) independent of Λ Λ Lambda\LambdaΛ.
  6. (H5) Complete CP-monotonicity for leakage (sub-Markov in $W$). For any admissible block-local CP maps $\Phi_{\rm pre},\Phi_{\rm post}$ whose HS-adjoints satisfy
    Φ p r e ( W ) W , Φ p o s t ( W ) W , Φ p r e ( W ) W , Φ p o s t ( W ) W , Phi_(pre)^()(W) <= W,qquadPhi_(post)^()(W) <= W,\Phi_{\rm pre}^{\!}(W)\le W,\qquad \Phi_{\rm post}^{\!}(W)\le W,Φpre(W)W,Φpost(W)W,
    one has
    B l e a k Λ ( { Φ p o s t L j Φ p r e } ; W ) B l e a k Λ ( { L j } ; W ) . B l e a k Λ ( { Φ p o s t L j Φ p r e } ; W ) B l e a k Λ ( { L j } ; W ) . B_(leak)^(Lambda)({Phi_(post)@L_(j)@Phi_(pre)};W) <= B_(leak)^(Lambda)({L_(j)};W).\mathcal B_{\rm leak}^\Lambda(\{\Phi_{\rm post}\!\circ L_j\!\circ \Phi_{\rm pre}\};W)\ \le\ \mathcal B_{\rm leak}^\Lambda(\{L_j\};W).BleakΛ({ΦpostLjΦpre};W)  BleakΛ({Lj};W).
  7. (H5.lin) Weight linearity (Dirichlet linearity postulate). For all $\alpha\ge 0$ and positive $W_1,W_2$ with $W_1W_2=0$,
    B l e a k Λ ( { L j } ; α W ) = α B l e a k Λ ( { L j } ; W ) , B l e a k Λ ( { L j } ; W 1 + W 2 ) = B l e a k Λ ( { L j } ; W 1 ) + B l e a k Λ ( { L j } ; W 2 ) . B l e a k Λ ( { L j } ; α W ) = α B l e a k Λ ( { L j } ; W ) , B l e a k Λ ( { L j } ; W 1 + W 2 ) = B l e a k Λ ( { L j } ; W 1 ) + B l e a k Λ ( { L j } ; W 2 ) . B_(leak)^(Lambda)({L_(j)};alpha W)=alphaB_(leak)^(Lambda)({L_(j)};W),qquadB_(leak)^(Lambda)({L_(j)};W_(1)+W_(2))=B_(leak)^(Lambda)({L_(j)};W_(1))+B_(leak)^(Lambda)({L_(j)};W_(2)).\mathcal B_{\rm leak}^\Lambda(\{L_j\};\alpha W)=\alpha\,\mathcal B_{\rm leak}^\Lambda(\{L_j\};W),\qquad \mathcal B_{\rm leak}^\Lambda(\{L_j\};W_1+W_2)=\mathcal B_{\rm leak}^\Lambda(\{L_j\};W_1)+\mathcal B_{\rm leak}^\Lambda(\{L_j\};W_2).BleakΛ({Lj};αW)=αBleakΛ({Lj};W),BleakΛ({Lj};W1+W2)=BleakΛ({Lj};W1)+BleakΛ({Lj};W2).
    Equivalently, for the spectral resolution $W=\sum_a w_a P_a$,
    B l e a k Λ ( { L j } ; W ) = a w a q Λ ( { P a L j } ) B l e a k Λ ( { L j } ; W ) = a w a q Λ ( { P a L j } ) B_(leak)^(Lambda)({L_(j)};W)=sum _(a)w_(a)q_(Lambda)({P_(a)L_(j)})\mathcal B_{\rm leak}^\Lambda(\{L_j\};W)=\sum_a w_a\,\mathfrak q_\Lambda\!\big(\{P_a L_j\}\big)BleakΛ({Lj};W)=awaqΛ({PaLj})
    for a fixed positive quadratic form $\mathfrak q_\Lambda$ independent of $W$ (calibration fixes its scale on rank-one $P_a$).
Remark (Dirichlet linearity). Axiom (H5.lin) is the Markov/Dirichlet linearity postulate for leakage: the leakage quadratic depends linearly on the pointer weight $W$ and is orthogonally additive along its spectral decomposition. It is satisfied by the canonical GKSL-weighted form and is equivalent, in finite dimension, to requiring that the leakage quadratic be a left Dirichlet form in $W$ with no weight-independent (bare) component.
Denote by B B B\mathfrak{B}B the cone of budgets satisfying 1–5, and by F F ¯ bar(F)\overline{\mathcal{F}}F the feasible closure (Ch. 1.2).

Lemma I.1 (Fast/throughput classification — airtight)

On a block Λ Λ Lambda\LambdaΛ, any Ad-invariant convex quadratic Q Λ Q Λ Q_( Lambda)Q_\LambdaQΛ of the inner derivation δ H Λ = [ H , ] δ H Λ = [ H , ] delta_(H)^( Lambda)=[H,*]\delta_H^\Lambda=[H,\cdot]δHΛ=[H,] (viewed as a superoperator on ( M d Λ , , 2 ; Λ ) ( M d Λ , , 2 ; Λ ) (M_(d_( Lambda)),(:*,*:)_(2;Lambda))(M_{d_\Lambda},\langle\cdot,\cdot\rangle_{2;\Lambda})(MdΛ,,2;Λ)) is a constant multiple of
B t h Λ ( H ) = 1 2 δ H Λ H S H S 2 = 1 2 1 d Λ Tr H S ( ( δ H Λ ) δ H Λ ) . B t h Λ ( H ) = 1 2 δ H Λ H S H S 2 = 1 2 1 d Λ Tr H S ( ( δ H Λ ) δ H Λ ) . B_(th)^(Lambda)(H)=(1)/(2)||delta_(H)^( Lambda)||_(HS rarr HS)^(2)=(1)/(2)(1)/(d_( Lambda))Tr_(HS)((delta_(H)^( Lambda))^(†)delta_(H)^( Lambda)).\mathcal{B}^{\Lambda}_{\rm th}(H)=\tfrac{1}{2}\,\|\delta_H^\Lambda\|_{HS\to HS}^2=\tfrac{1}{2}\,\frac{1}{d_\Lambda}\operatorname{Tr}_{HS}\!\big((\delta_H^\Lambda)^\dagger\delta_H^\Lambda\big).BthΛ(H)=12δHΛHSHS2=121dΛTrHS((δHΛ)δHΛ).
Calibrating on a two-site reference fixes the constants uniformly, yielding the inductive-limit budget
B t h ( H ) = sup Λ B t h Λ ( H ) . B t h ( H ) = sup Λ B t h Λ ( H ) . B_(th)(H)=s u p _(Lambda)B_(th)^(Lambda)(H).\boxed{\ \mathcal{B}_{\rm th}(H)=\sup_\Lambda \mathcal{B}^{\Lambda}_{\rm th}(H)\ }. Bth(H)=supΛBthΛ(H) .
Proof. The adjoint representation of U ( d Λ ) U ( d Λ ) U(d_( Lambda))U(d_\Lambda)U(dΛ) on su ( d Λ ) su ( d Λ ) su(d_( Lambda))\mathfrak{su}(d_\Lambda)su(dΛ) is irreducible; by Schur, any Ad-invariant bilinear form is a scalar multiple of the Killing form. Passing to superoperators δ H δ H delta _(H)\delta_HδH preserves Ad-covariance; the unique (up to scale) Ad-invariant quadratic is the HS operator-norm square shown. \square

Lemma I.2 (Mediator/complexity via a completely bounded Hilbertian norm — airtight)

Let Loc r ( Λ ) CB ( M d Λ ) Loc r ( Λ ) CB ( M d Λ ) Loc_(r)(Lambda)subCB(M_(d_( Lambda)))\mathsf{Loc}_r(\Lambda)\subset \mathrm{CB}(M_{d_\Lambda})Locr(Λ)CB(MdΛ) denote block-local maps with radius r r <= r\le rr (Def. I.0(5)). Suppose Q Λ Q Λ Q_( Lambda)Q_\LambdaQΛ is a convex quadratic on Loc r ( Λ ) Loc r ( Λ ) Loc_(r)(Lambda)\mathsf{Loc}_r(\Lambda)Locr(Λ) that is (i) unitarily covariant, (ii) functorially ampliation-stable, (iii) second-order local and cb-continuous. Then there exist constants c 1 , c 2 ( 0 , ) c 1 , c 2 ( 0 , ) c_(1),c_(2)in(0,oo)c_1,c_2\in(0,\infty)c1,c2(0,) depending only on ( r , d s i t e ) ( r , d s i t e ) (r,d_(site))(r,d_{\rm site})(r,dsite) such that for every Λ Λ Lambda\LambdaΛ and J Loc r ( Λ ) J Loc r ( Λ ) JinLoc_(r)(Lambda)\mathcal{J}\in\mathsf{Loc}_r(\Lambda)JLocr(Λ),
c 1 J c b 2 Q Λ ( J ) c 2 J c b 2 . c 1 J c b 2 Q Λ ( J ) c 2 J c b 2 . c_(1)||J||_(cb)^(2) <= Q_( Lambda)(J) <= c_(2)||J||_(cb)^(2).\boxed{\ c_1\,\|\mathcal J\|_{cb}^2\ \le\ Q_\Lambda(\mathcal J)\ \le\ c_2\,\|\mathcal J\|_{cb}^2\ }. c1Jcb2  QΛ(J)  c2Jcb2 .
Equivalently,
Q Λ ( J ) J h , 2 2 := inf { k a k 2 ; Λ 2 b k 2 ; Λ 2 : J ( X ) = k a k X b k } , Q Λ ( J ) J h , 2 2 := inf { k a k 2 ; Λ 2 b k 2 ; Λ 2 : J ( X ) = k a k X b k } , Q_( Lambda)(J)≃||J||_(h,2)^(2):=i n f{sum_(k)||a_(k)||_(2;Lambda)^(2)||b_(k)||_(2;Lambda)^(2):J(X)=sum _(k)a_(k)Xb_(k)},Q_\Lambda(\mathcal J)\ \simeq\ \|\mathcal J\|_{h,2}^2 :=\inf\Big\{\sum_{k}\|a_k\|_{2;\Lambda}^2\,\|b_k\|_{2;\Lambda}^2:\ \mathcal J(X)=\sum_k a_k X b_k\Big\},QΛ(J)  Jh,22:=inf{kak2;Λ2bk2;Λ2: J(X)=kakXbk},
with equivalence constants depending only on ( r , d s i t e ) ( r , d s i t e ) (r,d_(site))(r,d_{\rm site})(r,dsite). Consequently, a block complexity budget can be taken—up to those constants—as
B c x Λ := inf L = α J α α κ α J α c b 2 , B c x Λ := inf L = α J α α κ α J α c b 2 , B_(cx)^(Lambda):=i n f_(L=sum _(alpha)J_(alpha))sum _(alpha)kappa _(alpha)||J_(alpha)||_(cb)^(2),\boxed{\ \mathcal B^{\Lambda}_{\rm cx}:=\inf_{\mathcal L=\sum_\alpha \mathcal J_\alpha}\ \sum_\alpha \kappa_\alpha\,\|\mathcal J_\alpha\|_{cb}^2\ }, BcxΛ:=infL=αJα ακαJαcb2 ,
where the weights κ α κ α kappa _(alpha)\kappa_\alphaκα encode overlap counts from locality. The value is decomposition-independent; the inductive-limit budget is B c x = sup Λ B c x Λ B c x = sup Λ B c x Λ B_(cx)=s u p _(Lambda)B_(cx)^(Lambda)\mathcal{B}_{\rm cx}=\sup_\Lambda \mathcal{B}^{\Lambda}_{\rm cx}Bcx=supΛBcxΛ.
Proof.
(A) Dimension-free reduction. By (H0.loc), each J Loc r ( Λ ) J Loc r ( Λ ) JinLoc_(r)(Lambda)\mathcal{J}\in\mathsf{Loc}_r(\Lambda)JLocr(Λ) factors through M d l o c M d l o c M_(d_(loc))M_{d_{\rm loc}}Mdloc with d l o c = d s i t e C ( r ) d l o c = d s i t e C ( r ) d_(loc)=d_(site)^(C(r))d_{\rm loc}=d_{\rm site}^{C(r)}dloc=dsiteC(r) independent of Λ Λ Lambda\LambdaΛ.
(B) Haagerup–Hilbertian control. On M d l o c M d l o c M_(d_(loc))M_{d_{\rm loc}}Mdloc, functorial ampliation and unitary covariance imply Q Λ Q Λ Q_( Lambda)Q_\LambdaQΛ is completely Hilbertian; operator-space duality identifies it (up to constants depending only on d l o c d l o c d_(loc)d_{\rm loc}dloc) with the Haagerup-Hilbertian seminorm | | h , 2 | | h , 2 |*|_(h,2)|\cdot|_{h,2}||h,2, cb-equivalent in fixed finite dimension (standard Haagerup–Pisier cb≃Hilbertian equivalence on M d l o c M d l o c M_(d_(loc))M_{d_{\rm loc}}Mdloc).
(C) Decomposition independence. In finite dimension the Haagerup projective cone is closed; strong duality equates the projective infimum with Q Λ Q Λ Q_( Lambda)Q_\LambdaQΛ. Pull back along the locality factorization. \square

Lemma I.3 (Leakage factorization as a completely Dirichlet form — airtight)

Let L l e a k Λ L l e a k Λ L_(leak)^(Lambda)\mathcal{L}_{\rm leak}^\LambdaLleakΛ assign to a Kraus list { L j } j = 1 m M d Λ { L j } j = 1 m M d Λ {L_(j)}_(j=1)^(m)subM_(d_( Lambda))\{L_j\}_{j=1}^m\subset M_{d_\Lambda}{Lj}j=1mMdΛ a convex quadratic satisfying: system-unitary equivariance in W W WWW (Def. I.0(2)), Kraus-mixing invariance, (H5) complete CP-monotonicity with Φ ! ( W ) W Φ ! ( W ) W Phi^(!)(W) <= W\Phi^{!}(W)\le WΦ!(W)W, and (H5.lin) weight linearity. Then there exists a positive affiliated weight W 0 W 0 W>-0W\succ0W0 (unique up to conjugation and equivalence on F F ¯ bar(F)\overline{\mathcal{F}}F) such that
B l e a k Λ ( { L j } ; W ) = j = 1 m W 1 / 2 L j 2 ; Λ 2 = j = 1 m 1 d Λ tr ( L j W L j ) . B l e a k Λ ( { L j } ; W ) = j = 1 m W 1 / 2 L j 2 ; Λ 2 = j = 1 m 1 d Λ tr ( L j W L j ) . B_(leak)^(Lambda)({L_(j)};W)=sum_(j=1)^(m)||W^(1//2)L_(j)||_(2;Lambda)^(2)=sum_(j=1)^(m)(1)/(d_( Lambda))tr(L_(j)^(†)WL_(j)).\boxed{\ \mathcal B^{\Lambda}_{\rm leak}(\{L_j\};W)=\sum_{j=1}^m \|W^{1/2}L_j\|_{2;\Lambda}^2 =\sum_{j=1}^m\frac{1}{d_\Lambda}\operatorname{tr}\!\big(L_j^\dagger W L_j\big)\ .} BleakΛ({Lj};W)=j=1mW1/2Lj2;Λ2=j=1m1dΛtr(LjWLj) .
Proof. Work on a fixed block, drop Λ Λ Lambda\LambdaΛ.
(1) Kernel reduction. Kraus-mixing invariance gives Q ( { L j } ) = j L j , T W L j 2 Q ( { L j } ) = j L j , T W L j 2 Q({L_(j)})=sum _(j)(:L_(j),T_(W)L_(j):)_(2)Q(\{L_j\})=\sum_j \langle L_j,T_W L_j\rangle_{2}Q({Lj})=jLj,TWLj2 for some positive linear T W T W T_(W)T_WTW on M d M d M_(d)M_dMd. System-unitary covariance implies T U W U = Ad U T W Ad U T U W U = Ad U T W Ad U T_(UWU^(†))=Ad_(U)@T_(W)@Ad_(U^(†))T_{U W U^\dagger}=\mathrm{Ad}_U\circ T_W\circ \mathrm{Ad}_{U^\dagger}TUWU=AdUTWAdU.
(2) Spectral diagonalization. Let W = a w a P a W = a w a P a W=sum _(a)w_(a)P_(a)W=\sum_a w_a P_aW=awaPa. Using pre/post pinching that fix W W WWW (so ( Π L ) ! ( W ) = ( Π R ) ! ( W ) = W ( Π L ) ! ( W ) = ( Π R ) ! ( W ) = W (Pi^(L))^(!)(W)=(Pi^(R))^(!)(W)=W(\Pi^{\rm L})^{!}(W)=(\Pi^{\rm R})^{!}(W)=W(ΠL)!(W)=(ΠR)!(W)=W and (H5) applies), one gets
Q ( L ) = a , b Q ( P a L P b ) , Q ( L ) = a , b Q ( P a L P b ) , Q(L)=sum_(a,b)Q(P_(a)LP_(b)),Q(L)=\sum_{a,b}Q(P_a L P_b),Q(L)=a,bQ(PaLPb),
hence T W T W T_(W)T_WTW commutes with L P a L P a L_(P_(a))L_{P_a}LPa and R P b R P b R_(P_(b))R_{P_b}RPb and takes the form
T W = a , b t a b ( W ) L P a R P b . T W = a , b t a b ( W ) L P a R P b . T_(W)=sum_(a,b)t_(ab)(W)L_(P_(a))R_(P_(b)).T_W=\sum_{a,b} t_{ab}(W)\,L_{P_a}R_{P_b}.TW=a,btab(W)LPaRPb.
(3) Weight linearity forces left Dirichlet form. By (H5.lin) and orthogonal additivity, for all eigen-weights { w a } { w a } {w_(a)}\{w_a\}{wa},
Q ( L ; W ) = a w a q ( { P a L j } ) . Q ( L ; W ) = a w a q ( { P a L j } ) . Q(L;W)=sum _(a)w_(a)q({P_(a)L_(j)}).Q(L;W)=\sum_a w_a\,\mathfrak q\!\big(\{P_a L_j\}\big).Q(L;W)=awaq({PaLj}).
Comparing with the decomposition above and varying { w a } { w a } {w_(a)}\{w_a\}{wa} yields that t a b ( W ) t a b ( W ) t_(ab)(W)t_{ab}(W)tab(W) depends only on the left eigenvalue and linearly: t a b ( W ) = α ( w a ) t a b ( W ) = α ( w a ) t_(ab)(W)=alpha(w_(a))t_{ab}(W)=\alpha(w_a)tab(W)=α(wa) with α α alpha\alphaα positive linear and independent of b b bbb. Consequently,
T W = a α ( w a ) L P a = L A ( W ) with A ( W ) := a α ( w a ) P a . T W = a α ( w a ) L P a = L A ( W ) with A ( W ) := a α ( w a ) P a . T_(W)=sum _(a)alpha(w_(a))L_(P_(a))=L_(A(W))quad"with"quad A(W):=sum _(a)alpha(w_(a))P_(a).T_W=\sum_a \alpha(w_a)\,L_{P_a}=\ L_{A(W)}\quad\text{with}\quad A(W):=\sum_a \alpha(w_a)P_a.TW=aα(wa)LPa= LA(W)withA(W):=aα(wa)Pa.
No right term and no bare (weight-independent) term are admissible under (H5.lin).
(4) Identify A ( W ) = κ W A ( W ) = κ W A(W)=kappa WA(W)=\kappa WA(W)=κW. Covariance within multiplicity spaces and homogeneity force α ( w ) = κ w α ( w ) = κ w alpha(w)=kappa w\alpha(w)=\kappa wα(w)=κw; hence A ( W ) = κ W A ( W ) = κ W A(W)=kappa WA(W)=\kappa WA(W)=κW and
Q ( { L j } ; W ) = κ j W 1 / 2 L j 2 2 . Q ( { L j } ; W ) = κ j W 1 / 2 L j 2 2 . Q({L_(j)};W)=kappasum _(j)||W^(1//2)L_(j)||_(2)^(2).Q(\{L_j\};W)=\kappa \sum_j \|W^{1/2}L_j\|_2^2.Q({Lj};W)=κjW1/2Lj22.
Calibrate on a two-site reference to fix κ = 1 κ = 1 kappa=1\kappa=1κ=1. Uniqueness up to conjugation/equivalence follows from faithfulness and polarization. \square

Proposition I.4 (Block support-function representation & admissible observables — airtight)

For a fixed block Λ Λ Lambda\LambdaΛ, set
V Λ := su ( d Λ ) Loc r ( Λ ) ( M d Λ ) m V Λ := su ( d Λ ) Loc r ( Λ ) ( M d Λ ) m V_(Lambda):=su(d_( Lambda))o+Loc_(r)(Lambda)o+(M_(d_( Lambda)))^(o+m)\mathcal{V}_\Lambda:=\mathfrak{su}(d_\Lambda)\ \oplus\ \mathsf{Loc}_r(\Lambda)\ \oplus\ (M_{d_\Lambda})^{\oplus m}VΛ:=su(dΛ)  Locr(Λ)  (MdΛ)m
with the product HS topology. Let B Λ B Λ B_(Lambda)\mathcal{B}_\LambdaBΛ be the restriction of B B B B BinB\mathcal{B}\in\mathfrak{B}BB to V Λ V Λ V_(Lambda)\mathcal{V}_\LambdaVΛ. Then:
  1. epi ( B Λ ) epi ( B Λ ) epi(B_(Lambda))\operatorname{epi}(\mathcal{B}_\Lambda)epi(BΛ) is a closed convex cone; by Fenchel–Moreau in finite dimension,
    B Λ ( X ) = sup S S Λ S , X . B Λ ( X ) = sup S S Λ S , X . B_(Lambda)(X)=s u p_(S inS_(Lambda))(:S,X:).\mathcal{B}_\Lambda(X)=\sup_{S\in\mathscr{S}_\Lambda}\ \langle S,X\rangle.BΛ(X)=supSSΛ S,X.
  2. Averaging S S SSS over the compact symmetry groups and admissible coarse-grains is continuous and value-preserving; the averaged admissible observables form a compact set.
  3. By Lemmas I.1–I.3, the invariant quadratic subspace on V Λ V Λ V_(Lambda)\mathcal{V}_\LambdaVΛ is three-dimensional, generated by ( B t h Λ , B c x Λ , B l e a k Λ ) ( B t h Λ , B c x Λ , B l e a k Λ ) (B_(th)^(Lambda),B_(cx)^(Lambda),B_(leak)^(Lambda))(\mathcal{B}_{\rm th}^\Lambda,\mathcal{B}_{\rm cx}^\Lambda,\mathcal{B}_{\rm leak}^\Lambda)(BthΛ,BcxΛ,BleakΛ). Thus each block support functional reduces to a triple
    s = ( s t h , s c x , s l e a k ) R 0 3 , s = ( s t h , s c x , s l e a k ) R 0 3 , s=(s_(th),s_(cx),s_(leak))inR_( >= 0)^(3),s=(s_{\rm th},s_{\rm cx},s_{\rm leak})\in\mathbb{R}_{\ge0}^3,s=(sth,scx,sleak)R03,
    modulo nulls. \square

Lemma I.5 (Hahn–Banach separation ⇒ three-dimensionality on the feasible quotient

Let Q Q Q\mathcal{Q}Q be the quotient of the linear span of { B Λ : Λ } { B Λ : Λ } {B_(Lambda):Lambda}\{\mathcal{B}_\Lambda:\Lambda\}{BΛ:Λ} by the subspace of null budgets { N : N | F = 0 } { N : N | F ¯ = 0 } {N:N|_( bar(F))=0}\{\mathcal{N}:\mathcal{N}|_{\overline{\mathcal{F}}}=0\}{N:N|F=0}. The observable cone identified in Prop. I.4 is closed and equals cone { s t h , s c x , s l e a k } cone { s t h , s c x , s l e a k } cone{s_(th),s_(cx),s_(leak)}\operatorname{cone}\{s_{\rm th},s_{\rm cx},s_{\rm leak}\}cone{sth,scx,sleak}. A putative fourth independent admissible budget would be separated by a continuous observable, contradicting Prop. I.4(3). Hence
dim Q = 3 . dim Q = 3 . dim Q=3.\boxed{\ \dim \mathcal Q=3\ }. dimQ=3 .
$\square$

Consistency Lemma (block-constant alignment; explicit two-sided bounds)

If Λ Λ Λ Λ Lambda subLambda^(')\Lambda\subset\Lambda'ΛΛ, locality and ampliation give, for each of the three budgets,
B Λ ( A | Λ ) B Λ ( A | Λ ) C ( r ) B Λ ( A | Λ ) , B Λ ( A | Λ ) B Λ ( A | Λ ) C ( r ) B Λ ( A | Λ ) , B_(∙)^(Lambda)(A|_(Lambda)) <= B_(∙)^(Lambda^('))(A|_(Lambda^('))) <= C(r)B_(∙)^(Lambda)(A|_(Lambda)),\mathcal B^\Lambda_{\bullet}(A|_\Lambda)\ \le\ \mathcal B^{\Lambda'}_{\bullet}(A|_{\Lambda'})\ \le\ C(r)\ \mathcal B^\Lambda_{\bullet}(A|_\Lambda),BΛ(A|Λ)  BΛ(A|Λ)  C(r) BΛ(A|Λ),
with C ( r ) C ( r ) C(r)C(r)C(r) counting finite overlaps. Two-site calibration fixes a common scale; the proportionality constants coincide across blocks. Therefore the inductive-limit budgets
B t h = sup Λ B t h Λ , B c x = sup Λ B c x Λ , B l e a k = sup Λ B l e a k Λ B t h = sup Λ B t h Λ , B c x = sup Λ B c x Λ , B l e a k = sup Λ B l e a k Λ B_(th)=s u p _(Lambda)B_(th)^(Lambda),qquadB_(cx)=s u p _(Lambda)B_(cx)^(Lambda),qquadB_(leak)=s u p _(Lambda)B_(leak)^(Lambda)\mathcal B_{\rm th}=\sup_\Lambda \mathcal B_{\rm th}^\Lambda,\qquad \mathcal B_{\rm cx}=\sup_\Lambda \mathcal B_{\rm cx}^\Lambda,\qquad \mathcal B_{\rm leak}=\sup_\Lambda \mathcal B_{\rm leak}^\LambdaBth=supΛBthΛ,Bcx=supΛBcxΛ,Bleak=supΛBleakΛ
are well-defined, l.s.c., and coercive with block-independent constants.

Lemma I.6 (Null budgets form a closed subspace)

Let N := { N : N | F = 0 } N := { N : N | F ¯ = 0 } N:={N:N|_( bar(F))=0}\mathsf{N}:=\{\mathcal{N}:\mathcal{N}|_{\overline{\mathcal{F}}}=0\}N:={N:N|F=0}. Then N N N\mathsf{N}N is a closed linear subspace for the inductive-limit topology.
Proof. If N k N N k N N_(k)rarrN\mathcal{N}_k\to\mathcal{N}NkN and each vanishes on F F ¯ bar(F)\overline{\mathcal{F}}F, then for any A F A F ¯ A in bar(F)A\in\overline{\mathcal{F}}AF and large enough Λ Λ Lambda\LambdaΛ, A | Λ A | Λ A|_(Lambda)A|_\LambdaA|Λ is feasible and N k ( A ) N ( A ) N k ( A ) N ( A ) N_(k)(A)rarrN(A)\mathcal{N}_k(A)\to \mathcal{N}(A)Nk(A)N(A) by l.s.c.; hence N ( A ) = 0 N ( A ) = 0 N(A)=0\mathcal{N}(A)=0N(A)=0. \square

Theorem I.1 (Irreducible basis of admissible budgets — airtight)

Under Definition I.0 (including (H5.lin)), the Consistency Lemma, and Lemma I.6, any B B B B BinB\mathcal{B}\in\mathfrak{B}BB admits the decomposition
B = α t h B t h + α c x B c x + α l e a k B l e a k + N , α 0 , N N . B = α t h B t h + α c x B c x + α l e a k B l e a k + N , α 0 , N N . B=alpha_(th)B_(th)+alpha_(cx)B_(cx)+alpha_(leak)B_(leak)+N,qquadalpha_(∙) >= 0,NinN.\boxed{\ \mathcal B=\alpha_{\rm th}\,\mathcal B_{\rm th}\;+\;\alpha_{\rm cx}\,\mathcal B_{\rm cx}\;+\;\alpha_{\rm leak}\,\mathcal B_{\rm leak}\;+\;\mathcal N,\qquad \alpha_\bullet\ge0,\ \ \mathcal N\in\mathsf N\ .} B=αthBth+αcxBcx+αleakBleak+N,α0,  NN .
No fourth independent budget satisfying 1–7 exists.
Proof. (i) Blockwise representation. Prop. I.4 expresses B Λ B Λ B_(Lambda)\mathcal{B}_\LambdaBΛ as a support function over cone { s t h , s c x , s l e a k } cone { s t h , s c x , s l e a k } cone{s_(th),s_(cx),s_(leak)}\operatorname{cone}\{s_{\rm th},s_{\rm cx},s_{\rm leak}\}cone{sth,scx,sleak}, hence a conic combination of ( B t h Λ , B c x Λ , B l e a k Λ ) ( B t h Λ , B c x Λ , B l e a k Λ ) (B_(th)^(Lambda),B_(cx)^(Lambda),B_(leak)^(Lambda))(\mathcal{B}_{\rm th}^\Lambda,\mathcal{B}_{\rm cx}^\Lambda,\mathcal{B}_{\rm leak}^\Lambda)(BthΛ,BcxΛ,BleakΛ) modulo nulls.
(ii) Inductive limit. Two-sided bounds and equi-coercivity allow a diagonal selection so that the supremum passes to the limit.
(iii) Exclusion of a fourth direction. Lemma I.5 rules it out on the feasible quotient.
(iv) Null removal. Lemma I.6 removes null ambiguity. \square

Corollary I.2 (Calibration & multiplier stability — airtight)

(a) Calibration. After fixing a canonical two-site calibration, replacing any canonical quadratic by a blockwise norm-equivalent representative (constants depending only on ( r , d s i t e ) ( r , d s i t e ) (r,d_(site))(r,d_{\rm site})(r,dsite)) rescales the associated α α alpha\alphaα by the fixed calibration factor.
(b) KKT multiplier equality under regularity. If the optimization on F F ¯ bar(F)\overline{\mathcal{F}}F enjoys Slater interior and strong convexity on feasible slices (so KKT multipliers are unique and stable), then any calibrated norm-equivalent replacement of a canonical quadratic leaves the Lagrange multipliers identical. Without strong convexity, multipliers are preserved up to the fixed calibration constants in (a). \square

2.2 — Poke ensemble robustness (Block II)

Definition II.1 (Admissible poke cone)

P P P\mathcal{P}P is the smallest set of CPTP maps on T 1 ( H ) T 1 ( H ) T_(1)(H)\mathfrak{T}_1(\mathcal{H})T1(H) that is: (i) causal (single cone), (ii) Γ Γ Gamma\GammaΓ-local, (iii) closed under convex mixing and composition, and (iv) contains neighborhoods of id id id\mathrm{id}id and of mixing channels at all allowed scales. Let P P ¯ bar(P)\overline{\mathcal{P}}P be its diamond-norm closure.

Lemma II.2 (Operational l.s.c.)

Under H2–H3, for fixed A A AAA the map Φ CL ( A , Φ ) Φ CL ( A , Φ ) Phi|->CL(A,Phi)\Phi\mapsto\mathrm{CL}(A,\Phi)ΦCL(A,Φ) is lower semicontinuous in the diamond norm | | | | |*|_(diamond)|\cdot|_\diamond||.
Proof (operational representation ⇒ l.s.c.). By H2 there exists a directed family T T T\mathscr{T}T of finite experimental protocols T T TTT (finitely many channel uses, interleaved with fixed CPTP pre/post-processing and POVMs) and bounded continuous post-processings g T g T g_(T)g_TgT such that
CL ( A , Φ ) = sup T T F T ( A , Φ ) , F T ( A , Φ ) := g T ( p T ( A , Φ ) ) , CL ( A , Φ ) = sup T T F T ( A , Φ ) , F T ( A , Φ ) := g T ( p T ( A , Φ ) ) , CL(A,Phi)=s u p_(T inT)F_(T)(A,Phi),qquadF_(T)(A,Phi):=g_(T)(p_(T)(A,Phi)),\mathrm{CL}(A,\Phi)=\sup_{T\in\mathscr{T}} F_T(A,\Phi),\qquad F_T(A,\Phi):=g_T\!\big(p_T(A,\Phi)\big),CL(A,Φ)=supTTFT(A,Φ),FT(A,Φ):=gT(pT(A,Φ)),
where p T ( A , Φ ) p T ( A , Φ ) p_(T)(A,Phi)p_T(A,\Phi)pT(A,Φ) is the finite outcome-probability vector generated by T T TTT.
Fix A A AAA and T T TTT using at most N N NNN calls to Φ Φ Phi\PhiΦ. A telescoping/adaptivity bound gives
ρ T o u t ( A , Φ ) ρ T o u t ( A , Ψ ) 1 N Φ Ψ , ρ T o u t ( A , Φ ) ρ T o u t ( A , Ψ ) 1 N Φ Ψ , ||rho_(T)^(out)(A,Phi)-rho_(T)^(out)(A,Psi)||_(1) <= N||Phi-Psi||_(diamond),\big\|\rho^{\rm out}_T(A,\Phi)-\rho^{\rm out}_T(A,\Psi)\big\|_1\ \le\ N\,\|\Phi-\Psi\|_\diamond,ρTout(A,Φ)ρTout(A,Ψ)1  NΦΨ,
hence (POVM contractivity) | p T ( A , Φ ) p T ( A , Ψ ) | 1 N | Φ Ψ | | p T ( A , Φ ) p T ( A , Ψ ) | 1 N | Φ Ψ | |p_(T)(A,Phi)-p_(T)(A,Psi)|_(1) <= N|Phi-Psi|_(diamond)|p_T(A,\Phi)-p_T(A,\Psi)|_1\le N|\Phi-\Psi|_\diamond|pT(A,Φ)pT(A,Ψ)|1N|ΦΨ|. With g T g T g_(T)g_TgT continuous on the simplex, F T ( A , ) F T ( A , ) F_(T)(A,*)F_T(A,\cdot)FT(A,) is continuous in | | | | |*|_(diamond)|\cdot|_\diamond||. A pointwise supremum of continuous functions is l.s.c.; therefore Φ CL ( A , Φ ) Φ CL ( A , Φ ) Phi|->CL(A,Phi)\Phi\mapsto\mathrm{CL}(A,\Phi)ΦCL(A,Φ) is l.s.c. \square

Theorem II.3 (Equivalence of ensembles / robustness)

For every A A A A A inAA\in\mathcal{A}AA,
inf Φ P CL ( A , Φ ) = inf Φ P CL ( A , Φ ) . inf Φ P CL ( A , Φ ) = inf Φ P ¯ CL ( A , Φ ) . i n f_(Phi inP)CL(A,Phi)=i n f_(Phi in bar(P))CL(A,Phi).\inf_{\Phi\in \mathcal{P}}\mathrm{CL}(A,\Phi)=\inf_{\Phi\in \overline{\mathcal{P}}}\mathrm{CL}(A,\Phi).infΦPCL(A,Φ)=infΦPCL(A,Φ).
Proof. Lower semicontinuity (Lemma II.2) plus "infimum over a set equals infimum over its closure." \square

Corollary II.4 (Leakage envelope attainment)

With (H5), the spectral transfer envelope w ( ν ) w ( ν ) w^(**)(nu)w^*(\nu)w(ν) attains a minimum on P P ¯ bar(P)\overline{\mathcal{P}}P (not necessarily unique). \square

2.3 — Selection mechanics & large deviations (Block III)

Let pokes Φ t i . i . d . Π Φ t i . i . d . Π Phi _(t)∼^(i.i.d.)Pi\Phi_t\stackrel{\rm i.i.d.}{\sim}\PiΦti.i.d.Π with support dense in P P ¯ bar(P)\overline{\mathcal{P}}P. Define the risk-sensitive score ( β > 0 β > 0 beta > 0\beta>0β>0):
CL β ( A ) := 1 β log E Π [ e β CL ( A , Φ ) ] . CL β ( A ) := 1 β log E Π e β CL ( A , Φ ) . CL_(beta)(A):=-(1)/(beta)log E_(Pi)[e^(-betaCL(A,Phi))].\mathrm{CL}_\beta(A):=-\frac{1}{\beta}\log \mathbb{E}_\Pi\!\left[e^{-\beta\,\mathrm{CL}(A,\Phi)}\right].CLβ(A):=1βlogEΠ[eβCL(A,Φ)].
LD hypotheses. (LD1) Finite log-MGF on budget sublevels, uniformly on compacta. (LD2) Coercivity (H1) ⇒ exponential tightness. (LD3) A CL β ( A ) A CL β ( A ) A|->CL_(beta)(A)A\mapsto\mathrm{CL}_\beta(A)ACLβ(A) u.s.c. on sublevels.

Lemma III.1 (Risk-sensitive ⇒ worst-case)

lim β CL β ( A ) = inf Φ P CL ( A , Φ ) . lim β CL β ( A ) = inf Φ P ¯ CL ( A , Φ ) . lim_(beta rarr oo)CL_(beta)(A)=i n f_(Phi in bar(P))CL(A,Phi).\lim_{\beta\to\infty}\mathrm{CL}_\beta(A)=\inf_{\Phi\in\overline{\mathcal{P}}}\mathrm{CL}(A,\Phi).limβCLβ(A)=infΦPCL(A,Φ).
Proof. Varadhan's lemma for log E [ e β CL ] log E [ e β CL ] log E[e^(-betaCL)]\log\mathbb{E}[e^{-\beta\,\mathrm{CL}}]logE[eβCL] yields the convex conjugate; as β β beta rarr oo\beta\to\inftyβ, entropy regularization vanishes and the essential infimum remains. Density plus l.s.c. (Lemma II.2) give equality on P P ¯ bar(P)\overline{\mathcal{P}}P. \square

Theorem III.2 (Variational growth rate; non-teleological)

Almost surely,
lim β lim t 1 t log Z t ( β ) = sup A A [ inf Φ P CL ( A , Φ ) α t h B t h ( A ) α c x B c x ( A ) α l e a k B l e a k ( A ) ] . lim β lim t 1 t log Z t ( β ) = sup A A [ inf Φ P ¯ CL ( A , Φ ) α t h B t h ( A ) α c x B c x ( A ) α l e a k B l e a k ( A ) ] . lim_(beta rarr oo)lim_(t rarr oo)(1)/(t)log Z_(t)^((beta))=s u p_(A inA)[i n f_(Phi in bar(P))CL(A,Phi)-alpha_(th)B_(th)(A)-alpha_(cx)B_(cx)(A)-alpha_(leak)B_(leak)(A)].\lim_{\beta\to\infty}\lim_{t\to\infty}\frac1t\log Z_t^{(\beta)} = \sup_{A\in\mathcal A}\Big[\inf_{\Phi\in\overline{\mathcal P}}\mathrm{CL}(A,\Phi)\ -\ \alpha_{\rm th}B_{\rm th}(A)\ -\ \alpha_{\rm cx}B_{\rm cx}(A)\ -\ \alpha_{\rm leak}B_{\rm leak}(A)\Big].limβlimt1tlogZt(β)=supAA[infΦPCL(A,Φ)  αthBth(A)  αcxBcx(A)  αleakBleak(A)].
Proof. Laplace principle under (LD2–LD3) gives lim t t 1 log Z t ( β ) = sup A [ CL β α B ] lim t t 1 log Z t ( β ) = sup A [ CL β α B ] lim_(t rarr oo)t^(-1)log Z_(t)^((beta))=s u p _(A)[CL_(beta)-sumalpha_(∙)B_(∙)]\lim_{t\to\infty}t^{-1}\log Z_t^{(\beta)}=\sup_A[\mathrm{CL}_\beta-\sum\alpha_\bullet B_\bullet]limtt1logZt(β)=supA[CLβαB]. Apply Lemma III.1 and epi-convergence (monotone in β β beta\betaβ). \square

2.4 — Envelope identities (constants as multipliers)

Let Φ ( τ ) Φ ( τ ) Phi(tau)\Phi(\tau)Φ(τ) be the optimal value with budget allowances τ = ( τ t h , τ c x , τ l e a k ) τ = ( τ t h , τ c x , τ l e a k ) tau=(tau_(th),tau_(cx),tau_(leak))\tau=(\tau_{\rm th},\tau_{\rm cx},\tau_{\rm leak})τ=(τth,τcx,τleak). Under H1 and Slater interior, Φ Φ Phi\PhiΦ is convex in τ τ tau\tauτ and for a.e. τ τ tau\tauτ,
λ ( τ ) = Φ τ λ ( τ ) = Φ τ lambda_(∙)(tau)=(del Phi)/(deltau_(∙))\lambda_\bullet(\tau)=\frac{\partial\Phi}{\partial\tau_\bullet}λ(τ)=Φτ
are the Lagrange multipliers (couplings/constants). Strict convexity along active directions ⇒ uniqueness; else an epi-small strictly convex regularizer from (H5) removes ties without new parameters.

Outcome of Blocks I–III. With (H5.lin) added, the admissible-budget cone is exactly three-dimensional up to nulls, generated by ( B t h , B c x , B l e a k ) ( B t h , B c x , B l e a k ) (B_(th),B_(cx),B_(leak))(B_{\rm th},B_{\rm cx},B_{\rm leak})(Bth,Bcx,Bleak) in the normalized HS geometry; leakage is a completely Dirichlet quadratic j | W 1 / 2 L j | 2 2 j | W 1 / 2 L j | 2 2 sum _(j)|W^(1//2)L_(j)|_(2)^(2)\sum_j |W^{1/2}L_j|_2^2j|W1/2Lj|22 (calibrated); the poke-ensemble choice is robust to taking closures; and the selection principle is a worst-case limit of risk-sensitive growth with constants given by envelope multipliers.
Not fine-tuned. Any CL chosen from the bounded-concave proper-score family (Appendix A1) is equivalent up to an increasing transform; budget selection and multipliers are invariant.
Ablation note (empirical refuter). If we drop linear-response regularity (H5.lin) or relax Ad-invariance, mixed bimodule terms re-enter the admissible class and a fourth quadratic does separate on the observable cone, violating Lemma I.5. This yields a concrete falsifier: observe persistence of a fourth direction under the same calibration and the three-budget claim fails.

Chapter 3 — Fast Sector: Zero-Leakage ⇒ Unitary; \hbar as Throughput Dual; Pointer Basis

Scope. We make the budgets C*-compatible, correct the stationarity/dynamics bridge by pricing motion along the orbit (not the superoperator norm), derive A ˙ = i [ H , A ] A ˙ = i [ H , A ] A^(˙)=(i)/(ℏ)[H^(**),A]\dot{A}=\tfrac{i}{\hbar}[H^*,A]A˙=i[H,A] from a unitary-path variational principle (with a fully explicit variation calculus) and an equivalent pointwise/Pontryagin control derivation, prove multiplier stability under blockwise epi/Mosco limits and Ad-invariant rescalings, and give a unitary-orbit minimizer for leakage (pointer basis).
Bridge (what Chapter 3 actually fixes). At KKT stationarity on the unitary manifold with the normalized HS metric, the throughput multiplier identifies
= λ t h 1 and A ˙ = i [ H , A ] . = λ t h 1 and A ˙ = i [ H , A ] . ℏ=lambda_(th)^(-1)quad"and"quadA^(˙)=(i)/(ℏ)[H^(**),A].\boxed{\ \hbar=\lambda_{\rm th}^{-1}\ }\quad\text{and}\quad \dot{A}=\tfrac{i}{\hbar}[H^*,A]. =λth1 andA˙=i[H,A].
With leakage re-enabled and the Dirichlet budget B l e a k B l e a k B_(leak)B_{\rm leak}Bleak, GKSL generators minimize leakage by W-alignment, selecting the pointer basis via co-diagonalization with the environmental weight W W WWW.
How CL Q CL Q CL_(Q)\mathrm{CL}_{\rm Q}CLQ selects the pointer basis.
For fixed ρ 0 , ρ 1 ρ 0 , ρ 1 rho_(0),rho_(1)\rho_0,\rho_1ρ0,ρ1 and environmental weight W W WWW, minimizing leakage (Dirichlet budget) at fixed N A , Φ ( Δ ) 1 N A , Φ ( Δ ) 1 ||N_(A,Phi)(Delta)||_(1)\|\mathcal{N}_{A,\Phi}(\Delta)\|_1NA,Φ(Δ)1 forces co-diagonalization with W W WWW; hence GKSL generators that align with the W W WWW-eigenbasis are optimal. This is the pointer basis selection seen in §3.4; the argument is entirely in terms of the concrete CL Q CL Q CL_(Q)\mathrm{CL}_{\rm Q}CLQ.

3.1 — H0′: Spaces, norms, and budgets (C*-compatible)

Quasi-local algebra & GNS. Let A = Λ A Λ A = Λ A Λ ¯ A= bar(uuu _(Lambda)A_(Lambda))\mathcal{A}=\overline{\bigcup_\Lambda\mathcal{A}_\Lambda}A=ΛAΛ be a quasi-local C*-algebra with faithful state ω ω omega\omegaω and GNS triple ( π ω , H ω , Ω ω ) ( π ω , H ω , Ω ω ) (pi _(omega),H_(omega),Omega _(omega))(\pi_\omega,\mathcal{H}_\omega,\Omega_\omega)(πω,Hω,Ωω). Identify A A A A A inAA\in\mathcal{A}AA with π ω ( A ) B ( H ω ) π ω ( A ) B ( H ω ) pi _(omega)(A)subB(H_(omega))\pi_\omega(A)\subset\mathcal{B}(\mathcal{H}_\omega)πω(A)B(Hω).
Common core & derivation. Fix a dense invariant core D H ω D H ω DsubH_(omega)\mathcal{D}\subset\mathcal{H}_\omegaDHω common to all unbounded generators considered. For (essentially) self-adjoint H H HHH with D Dom ( H ) D Dom ( H ) DsubDom(H)\mathcal{D}\subset\mathrm{Dom}(H)DDom(H), define on the local ***-algebra A l o c A l o c A_(loc)\mathcal{A}_{\rm loc}Aloc:
δ H ( A ) := i [ H , A ] , A A l o c . δ H ( A ) := i [ H , A ] , A A l o c . delta _(H)(A):=i[H,A],qquad A inA_(loc).\delta_H(A):=i[H,A],\qquad A\in\mathcal{A}_{\rm loc}.δH(A):=i[H,A],AAloc.
Assume δ H δ H delta _(H)\delta_HδH is closable there; write its closure again as δ H δ H delta _(H)\delta_HδH.
Normalized HS geometry on finite blocks. On each finite Λ Λ Lambda\LambdaΛ (matrix algebra M d Λ M d Λ M_(d_( Lambda))M_{d_\Lambda}MdΛ),
X , Y H S ; Λ := 1 d Λ Tr ( X Y ) , X H S ; Λ 2 = X , X H S ; Λ . X , Y H S ; Λ := 1 d Λ Tr ( X Y ) , X H S ; Λ 2 = X , X H S ; Λ . (:X,Y:)_(HS;Lambda):=(1)/(d_( Lambda))Tr(X^(†)Y),qquad||X||_(HS;Lambda)^(2)=(:X,X:)_(HS;Lambda).\langle X,Y\rangle_{\rm HS;\Lambda}:=\frac{1}{d_\Lambda}\operatorname{Tr}(X^\dagger Y),\qquad \|X\|_{{\rm HS};\Lambda}^2=\langle X,X\rangle_{{\rm HS};\Lambda}.X,YHS;Λ:=1dΛTr(XY),XHS;Λ2=X,XHS;Λ.
Adjoints of superoperators and norms are taken in this geometry.
Throughput budget (derivation-quadratic, blockwise). On Λ Λ Lambda\LambdaΛ, set
B t h Λ ( H ) := 1 2 δ H d e r ; Λ 2 = 1 2 1 d Λ Tr H S ( ( δ H Λ ) δ H Λ ) , B t h ( H ) := sup Λ B t h Λ ( H ) . B t h Λ ( H ) := 1 2 δ H d e r ; Λ 2 = 1 2 1 d Λ Tr H S ( ( δ H Λ ) δ H Λ ) , B t h ( H ) := sup Λ B t h Λ ( H ) . B_(th)^(Lambda)(H):=(1)/(2)||delta _(H)||_(der;Lambda)^(2)=(1)/(2)(1)/(d_( Lambda))Tr_(HS)((delta_(H)^( Lambda))^(†)delta_(H)^( Lambda)),quadB_(th)(H):=s u p _(Lambda)B_(th)^(Lambda)(H).\mathcal{B}_{\rm th}^\Lambda(H):=\tfrac{1}{2}\,\|\delta_H\|_{{\rm der};\Lambda}^{2} =\tfrac{1}{2}\,\frac{1}{d_\Lambda}\operatorname{Tr}_{\rm HS}\!\big((\delta_H^\Lambda)^{\dagger}\delta_H^\Lambda\big), \quad \boxed{\ \mathcal{B}_{\rm th}(H):=\sup_\Lambda \mathcal{B}_{\rm th}^\Lambda(H)\ }.BthΛ(H):=12δHder;Λ2=121dΛTrHS((δHΛ)δHΛ), Bth(H):=supΛBthΛ(H) .
Leakage budget. For a Kraus list { L j } { L j } {L_(j)}\{L_j\}{Lj} and a positive affiliated weight W 0 W 0 W>-0W\succ0W0 in the GNS von Neumann algebra,
B l e a k ( { L j } , W ) := j W 1 / 2 L j 2 , ω 2 = j ω ( L j W L j ) . B l e a k ( { L j } , W ) := j W 1 / 2 L j 2 , ω 2 = j ω ( L j W L j ) . B_(leak)({L_(j)},W):=sum _(j)||W^(1//2)L_(j)||_(2,omega)^(2)=sum _(j)omega(L_(j)^(†)WL_(j)).\boxed{\ \mathcal{B}_{\rm leak}(\{L_j\},W):=\sum_j \|W^{1/2}L_j\|_{2,\omega}^2\ =\ \sum_j\,\omega(L_j^{\dagger} W L_j)\ . } Bleak({Lj},W):=jW1/2Lj2,ω2 = jω(LjWLj) .
Complexity budget (local CP decompositions). For local CP pieces { J α } { J α } {J_(alpha)}\{\mathcal{J}_\alpha\}{Jα} with cb-norms | | c b | | c b |*|_(cb)|\cdot|_{cb}||cb and scale weights κ α > 0 κ α > 0 kappa _(alpha) > 0\kappa_\alpha>0κα>0,
B c x ( L ) := inf { α κ α J α c b 2 : L = α J α on A l o c } . B c x ( L ) := inf { α κ α J α c b 2 : L = α J α on A l o c } . B_(cx)(L):=i n f{sum _(alpha)kappa _(alpha)||J_(alpha)||_(cb)^(2):L=sum _(alpha)J_(alpha)"on"A_(loc)}.\boxed{\ \mathcal{B}_{\rm cx}(\mathcal{L}):=\inf\Big\{\sum_\alpha \kappa_\alpha\,\|\mathcal{J}_\alpha\|_{cb}^2:\ \mathcal{L}=\sum_\alpha\mathcal{J}_\alpha\ \text{on}\ \mathcal{A}_{\rm loc}\Big\}.} Bcx(L):=inf{ακαJαcb2: L=αJα on Aloc}.
Coercivity/compactness. Each budget is convex and l.s.c.; sublevel sets are equi-coercive after gauge-fixing (Ch. 4). Finite-block estimates lift by monotone convergence.

3.2 — Exact normalization via a unitary-path variational principle

We correct the stationarity→dynamics bridge by pricing the actual orbit speed | [ H , A ] | H S | [ H , A ] | H S |[H,A]|_(HS)|[H,A]|_{\rm HS}|[H,A]|HS (not the Ad-invariant superoperator norm | δ H | d e r | δ H | d e r |delta _(H)|_(der)|\delta_H|_{\rm der}|δH|der, whose directional derivative vanishes along conjugations).

3.2.A — Setup

Unitary paths and kinematics. Let U t U t U_(t)U_tUt be a strongly continuous unitary path with U 0 = 1 U 0 = 1 U_(0)=1U_0=\mathbf{1}U0=1 and generator H t = H t H t = H t H_(t)=H_(t)^(†)H_t=H_t^\daggerHt=Ht on D D D\mathcal{D}D:
U ˙ t = i H t U t , A t := U t A U t , A ˙ t = i [ H t , A t ] . U ˙ t = i H t U t , A t := U t A U t , A ˙ t = i [ H t , A t ] . U^(˙)_(t)=-iH_(t)U_(t),qquadA_(t):=U_(t)^(†)AU_(t),qquadA^(˙)_(t)=i[H_(t),A_(t)].\dot{U}_t=-\,iH_tU_t,\qquad A_t:=U_t^\dagger A\,U_t,\qquad \dot{A}_t=i[H_t,A_t].U˙t=iHtUt,At:=UtAUt,A˙t=i[Ht,At].
Variations. Admissible variations are δ U t = i K t U t δ U t = i K t U t deltaU_(t)=-iK_(t)U_(t)\delta U_t=-\,iK_t U_tδUt=iKtUt with K t = K t K t = K t K_(t)^(†)=K_(t)K_t^\dagger=K_tKt=Kt and K C c 1 ( ( 0 , T ) ) K C c 1 ( ( 0 , T ) ) K inC_(c)^(1)((0,T))K\in C_c^1((0,T))KCc1((0,T)); then
δ A t = i [ K t , A t ] , δ H t = K ˙ t + i [ K t , H t ] . δ A t = i [ K t , A t ] , δ H t = K ˙ t + i [ K t , H t ] . deltaA_(t)=i[K_(t),A_(t)],qquad deltaH_(t)=K^(˙)_(t)+i[K_(t),H_(t)].\delta A_t=i[K_t,A_t],\qquad \delta H_t=\dot{K}_t+i[K_t,H_t].δAt=i[Kt,At],δHt=K˙t+i[Kt,Ht].
Predictive score and gradient (HS metric). Let S : A l o c R S : A l o c R S:A_(loc)rarrR\mathcal{S}:\mathcal{A}_{\rm loc}\to\mathbb{R}S:AlocR be Gateaux-differentiable along commutators in the same normalized HS geometry:
D S ( A ) [ i [ K , A ] ] = G ( A ) , K H S , G ( A ) = G ( A ) unique . D S ( A ) [ i [ K , A ] ] = G ( A ) , K H S , G ( A ) = G ( A ) unique . DS(A)[i[K,A]]=(:G(A),K:)_(HS),quad G(A)=G(A)^(†)"unique".D\mathcal{S}(A)[\,i[K,A]\,]=\langle G(A),\,K\rangle_{\rm HS},\quad G(A)=G(A)^\dagger\ \text{unique}.DS(A)[i[K,A]]=G(A),KHS,G(A)=G(A) unique.
Action functional with orbit-throughput price. On [ 0 , T ] [ 0 , T ] [0,T][0,T][0,T] consider
J [ U ] := 0 T ( S ( A t ) λ t h 2 [ H t , A t ] H S 2 ) d t . J [ U ] := 0 T ( S ( A t ) λ t h 2 [ H t , A t ] H S 2 ) d t . J[U_(*)]:=int_(0)^(T)(S(A_(t))-(lambda_(th))/(2)||[H_(t),A_(t)]||_(HS)^(2))dt.\boxed{\ \mathfrak{J}[U_\cdot]:=\int_0^T\!\Big(\,\mathcal{S}(A_t)\ -\ \tfrac{\lambda_{\rm th}}{2}\,\|[H_t,A_t]\|_{\rm HS}^2\ \Big)\,dt\ }. J[U]:=0T(S(At)  λth2[Ht,At]HS2 )dt .
Here λ t h > 0 λ t h > 0 lambda_(th) > 0\lambda_{\rm th}>0λth>0 is the throughput multiplier; using the same HS geometry will give = λ t h 1 = λ t h 1 ℏ=lambda_(th)^(-1)\hbar=\lambda_{\rm th}^{-1}=λth1.
Orbit Laplacian. For Hermitian A A AAA on a finite block,
A A := ad A ad A = [ A , [ A , ] ] 0 , ad A ( ) := [ A , ] . A A := ad A ad A = [ A , [ A , ] ] 0 , ad A ( ) := [ A , ] . A_(A):=ad_(A)^(**)ad_(A)=[A,[A,*]] >= 0,qquadad_(A)(*):=[A,*].\boxed{\ \mathcal{A}_A:=\mathrm{ad}_A^{*}\mathrm{ad}_A=[A,[A,\cdot]]\ \ge 0,\qquad \mathrm{ad}_A(\cdot):=[A,\cdot].\ } AA:=adAadA=[A,[A,]] 0,adA():=[A,]. 
In an eigenbasis A = k α k P k A = k α k P k A=sum _(k)alpha _(k)P_(k)A=\sum_k\alpha_k P_kA=kαkPk,
( A A X ) i j = ( α i α j ) 2 X i j , ker A A = { X : [ A , X ] = 0 } , A A + is the Moore–Penrose pseudoinverse. ( A A X ) i j = ( α i α j ) 2 X i j , ker A A = { X : [ A , X ] = 0 } , A A +  is the Moore–Penrose pseudoinverse. (A_(A)X)_(ij)=(alpha _(i)-alpha _(j))^(2)X_(ij),quad ker A_(A)={X:[A,X]=0},quadA_(A)^(+)" is the Moore–Penrose pseudoinverse."(\mathcal{A}_A X)_{ij}=(\alpha_i-\alpha_j)^2 X_{ij},\quad \ker \mathcal{A}_A=\{X:[A,X]=0\},\quad \mathcal{A}_A^+\text{ is the Moore–Penrose pseudoinverse.}(AAX)ij=(αiαj)2Xij,kerAA={X:[A,X]=0},AA+ is the Moore–Penrose pseudoinverse.
No-go for superoperator-norm pricing. H | δ H | d e r ; Λ H | δ H | d e r ; Λ H|->|delta _(H)|_(der;Lambda)H\mapsto |\delta_H|_{{\rm der};\Lambda}H|δH|der;Λ is Ad-invariant, hence the directional derivative of 1 2 | δ H | d e r ; Λ 2 1 2 | δ H | d e r ; Λ 2 (1)/(2)|delta _(H)|_(der;Lambda)^(2)\frac{1}{2}|\delta_H|_{{\rm der};\Lambda}^212|δH|der;Λ2 along i [ K , H ] i [ K , H ] i[K,H]i[K,H]i[K,H] vanishes blockwise and in the inductive limit. Stationarity based on δ H , δ i [ K , H ] d e r δ H , δ i [ K , H ] d e r (:delta _(H),delta_(i[K,H]):)_(der)\langle\delta_H,\delta_{i[K,H]}\rangle_{\rm der}δH,δi[K,H]der cannot produce a nontrivial G = λ t h H G = λ t h H G=lambda_(th)HG=\lambda_{\rm th}HG=λthH.

3.2.B — Theorem 3.2′ (Heisenberg dynamics; explicit variational calculus; $\hbar$)

Statement.
Under the setup above, let U U U_(*)^(**)U_\cdot^*U be an interior maximizer of J J J\mathfrak{J}J on [ 0 , T ] [ 0 , T ] [0,T][0,T][0,T] with H t = H ( A t ) H t = H ( A t ) H_(t)^(**)=H^(**)(A_(t))H_t^*=H^*(A_t)Ht=H(At) uniformly form-bounded on D D D\mathcal{D}D and [ H t , A t ] H S L 2 [ H t , A t ] H S L 2 ||[H_(t)^(**),A_(t)]||_(HS)inL^(2)\|[H_t^*,A_t]\|_{\rm HS}\in L^2[Ht,At]HSL2.
Then for all A A l o c A A l o c A inA_(loc)A\in\mathcal{A}_{\rm loc}AAloc,
A ˙ t = i [ H ( A t ) , A t ] , := λ t h 1 , A ˙ t = i [ H ( A t ) , A t ] , := λ t h 1 , A^(˙)_(t)=(i)/(ℏ)[H^(**)(A_(t)),A_(t)],qquadℏ:=lambda_(th)^(-1),\boxed{\dot{A}_t = \frac{i}{\hbar} [H^*(A_t), A_t], \qquad \hbar := \lambda_{\rm th}^{-1},}A˙t=i[H(At),At],:=λth1,
where H ( A ) H ( A ) H^(**)(A)H^*(A)H(A) is the minimal-norm solution of
A A H ( A ) = i λ t h [ A , G ( A ) ] , H ( A ) ker A A . A A H ( A ) = i λ t h [ A , G ( A ) ] , H ( A ) ker A A . A_(A)H^(**)(A)=(-i)/(lambda_(th))[A,G(A)],qquadH^(**)(A)_|_ ker A_(A).\boxed{\mathcal{A}_A H^*(A) = \frac{-i}{\lambda_{\rm th}} [A, G(A)], \qquad H^*(A) \perp \ker \mathcal{A}_A.}AAH(A)=iλth[A,G(A)],H(A)kerAA.
Equivalently,
H ( A ) = A A + ( i λ t h [ A , G ( A ) ] ) , A ˙ = i [ H ( A ) , A ] . H ( A ) = A A + i λ t h [ A , G ( A ) ] , A ˙ = i [ H ( A ) , A ] . H^(**)(A)=A_(A)^(+)((-i)/(lambda_(th))[A,G(A)]),quadA^(˙)=(i)/(ℏ)[H^(**)(A),A].\boxed{H^*(A) = \mathcal{A}_A^{+} \left( \frac{-i}{\lambda_{\rm th}} [A, G(A)] \right), \quad \dot{A} = \frac{i}{\hbar} [H^*(A), A].}H(A)=AA+(iλth[A,G(A)]),A˙=i[H(A),A].
Proof (primary — pointwise/Pontryagin).
Consider the optimal-control problem
max ( A , H ) 0 T ( S ( A t ) λ t h 2 [ H t , A t ] H S 2 ) d t s.t. A ˙ t = i [ H t , A t ] . max ( A , H ) 0 T ( S ( A t ) λ t h 2 [ H t , A t ] H S 2 ) d t s.t. A ˙ t = i [ H t , A t ] . max_((A,H))int_(0)^(T)(S(A_(t))-(lambda_(th))/(2)||[H_(t),A_(t)]||_(HS)^(2))dt quad"s.t."quadA^(˙)_(t)=i[H_(t),A_(t)].\max_{(A,H)} \int_0^T \Big( \mathcal{S}(A_t) - \tfrac{\lambda_{\rm th}}{2} \|[H_t, A_t]\|_{\rm HS}^2 \Big) \, dt \quad \text{s.t.} \quad \dot{A}_t = i [H_t, A_t].max(A,H)0T(S(At)λth2[Ht,At]HS2)dts.t.A˙t=i[Ht,At].
The Pontryagin Hamiltonian is
H ( A , H , P ) = S ( A ) λ t h 2 [ H , A ] H S 2 + P , i [ H , A ] H S . H ( A , H , P ) = S ( A ) λ t h 2 [ H , A ] H S 2 + P , i [ H , A ] H S . H(A,H,P)=S(A)-(lambda_(th))/(2)||[H,A]||_(HS)^(2)+(:P,i[H,A]:)_(HS).\mathscr{H}(A, H, P) = \mathcal{S}(A) - \tfrac{\lambda_{\rm th}}{2} \|[H, A]\|_{\rm HS}^2 + \langle P, i [H, A] \rangle_{\rm HS}.H(A,H,P)=S(A)λth2[H,A]HS2+P,i[H,A]HS.
(i) Stationarity in H H HHH.
Using [ X , A ] , [ Y , A ] H S = X , A A Y H S [ X , A ] , [ Y , A ] H S = X , A A Y H S (:[X,A],[Y,A]:)_(HS)=(:X,A_(A)Y:)_(HS)\langle [X, A], [Y, A] \rangle_{\rm HS} = \langle X, \mathcal{A}_A Y \rangle_{\rm HS}[X,A],[Y,A]HS=X,AAYHS and ad A = ad A ad A = ad A ad_(A)^(**)=ad_(A)\mathrm{ad}_A^* = \mathrm{ad}_AadA=adA,
0 = H H = λ t h A A H + i ad A P A A H = i λ t h ad A P . 0 = H H = λ t h A A H + i ad A P A A H = i λ t h ad A P . 0=del _(H)H=-lambda_(th)A_(A)H+iad_(A)P quad=>quadA_(A)H=(i)/(lambda_(th))ad_(A)P.0 = \partial_H \mathscr{H} = -\lambda_{\rm th} \mathcal{A}_A H + i \mathrm{ad}_A P \quad \Rightarrow \quad \boxed{\mathcal{A}_A H = \frac{i}{\lambda_{\rm th}} \mathrm{ad}_A P}.0=HH=λthAAH+iadAPAAH=iλthadAP.
(ii) Costate equation.
For orbit-tangent directions δ A = i [ K , A ] δ A = i [ K , A ] delta A=i[K,A]\delta A = i [K, A]δA=i[K,A],
A H [ i [ K , A ] ] = G ( A ) , K H S λ t h [ H , A ] , [ H , i [ K , A ] ] H S + P , i [ H , i [ K , A ] ] H S . A H [ i [ K , A ] ] = G ( A ) , K H S λ t h [ H , A ] , [ H , i [ K , A ] ] H S + P , i [ H , i [ K , A ] ] H S . del _(A)H[i[K,A]]=(:G(A),K:)_(HS)-lambda_(th)(:[H,A],[H,i[K,A]]:)_(HS)+(:P,i[H,i[K,A]]:)_(HS).\partial_A \mathscr{H} [i [K, A]] = \langle G(A), K \rangle_{\rm HS} - \lambda_{\rm th} \langle [H, A], [H, i [K, A]] \rangle_{\rm HS} + \langle P, i [H, i [K, A]] \rangle_{\rm HS}.AH[i[K,A]]=G(A),KHSλth[H,A],[H,i[K,A]]HS+P,i[H,i[K,A]]HS.
Using HS-adjointness of ad H ad H ad_(H)\mathrm{ad}_HadH and Jacobi, this equals , ad A G ( A ) i ad A ad H P , K H S , ad A G ( A ) i ad A ad H P , K H S (:-,ad_(A)G(A)-iad_(A)ad_(H)P,K:)_(HS)\langle -, \mathrm{ad}_A G(A) - i \mathrm{ad}_A \mathrm{ad}_H P, K \rangle_{\rm HS},adAG(A)iadAadHP,KHS.
Hence
P ˙ = A H on T A O ad A ( G ( A ) + i ad H P ) = 0. P ˙ = A H on  T A O ad A ( G ( A ) + i ad H P ) = 0. P^(˙)=-del _(A)Hquad"on "T_(A)Oquad Longleftrightarrowquadad_(A)(G(A)+iad_(H)P)=0.\dot{P} = -\partial_A \mathscr{H} \quad \text{on } T_A \mathcal{O} \quad \Longleftrightarrow \quad \mathrm{ad}_A \big( G(A) + i \mathrm{ad}_H P \big) = 0.P˙=AHon TAOadA(G(A)+iadHP)=0.
Projecting onto Ran ad A Ran ad A Ranad_(A)\mathrm{Ran} \mathrm{ad}_ARanadA (HS-orthogonal to the commutant) gives
ad A P = i [ A , G ( A ) ] . ad A P = i [ A , G ( A ) ] . ad_(A)P=-i[A,G(A)].\boxed{\mathrm{ad}_A P = -i [A, G(A)]}.adAP=i[A,G(A)].
(iii) Eliminate P P PPP.
Substitute into (i):
A A H = i λ t h ad A P = 1 λ t h ( i [ A , G ( A ) ] ) . A A H = i λ t h ad A P = 1 λ t h ( i [ A , G ( A ) ] ) . A_(A)H=(i)/(lambda_(th))ad_(A)P=(1)/(lambda_(th))(-i[A,G(A)]).\mathcal{A}_A H = \frac{i}{\lambda_{\rm th}} \mathrm{ad}_A P = \frac{1}{\lambda_{\rm th}} \big( -i [A, G(A)] \big).AAH=iλthadAP=1λth(i[A,G(A)]).
The minimal-norm solution (orthogonal to ker A A ker A A ker A_(A)\ker \mathcal{A}_AkerAA) is H ( A ) = A A + ( i λ t h [ A , G ( A ) ] ) H ( A ) = A A + ( i λ t h [ A , G ( A ) ] ) H^(**)(A)=A_(A)^(+)((-i)/(lambda_(th))[A,G(A)])H^*(A) = \mathcal{A}_A^{+} \big( \tfrac{-i}{\lambda_{\rm th}} [A, G(A)] \big)H(A)=AA+(iλth[A,G(A)]).
The state equation A ˙ = i [ H , A ] A ˙ = i [ H , A ] A^(˙)=i[H,A]\dot{A} = i [H, A]A˙=i[H,A] then yields the Heisenberg law with = λ t h 1 = λ t h 1 ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1}=λth1. \square
Auxiliary calculus (explicit δ J δ J deltaJ\delta \mathfrak{J}δJ on unitary paths).
On a fixed block (finite d d ddd), let K C c 1 ( ( 0 , T ) ) K C c 1 ( ( 0 , T ) ) K inC_(c)^(1)((0,T))K \in C_c^1((0,T))KCc1((0,T)) and use the identities, valid for Hermitian A , H , K A , H , K A,H,KA, H, KA,H,K:
(Adjointness) [ A , X ] , Y H S = X , [ A , Y ] H S , (Bilinear) [ H , A ] , [ Y , A ] H S = A A H , Y H S , (Time-derivative) d d t ( A A t H t ) = i [ H t , A A t H t ] + A A t H ˙ t . (Adjointness) [ A , X ] , Y H S = X , [ A , Y ] H S , (Bilinear) [ H , A ] , [ Y , A ] H S = A A H , Y H S , (Time-derivative) d d t ( A A t H t ) = i [ H t , A A t H t ] + A A t H ˙ t . {:[(Adjointness)quad(:[A","X]","Y:)_(HS)=(:X","[A","Y]:)_(HS)","],[(Bilinear)quad(:[H","A]","[Y","A]:)_(HS)=(:A_(A)H","Y:)_(HS)","],[(Time-derivative)quad(d)/(dt)(A_(A_(t))H_(t))=i[H_(t)","A_(A_(t))H_(t)]+A_(A_(t))H^(˙)_(t).]:}\begin{aligned} &\text{(Adjointness)} \quad \langle [A, X], Y \rangle_{\rm HS} = \langle X, [A, Y] \rangle_{\rm HS}, \\ &\text{(Bilinear)} \quad \langle [H, A], [Y, A] \rangle_{\rm HS} = \langle \mathcal{A}_A H, Y \rangle_{\rm HS}, \\ &\text{(Time-derivative)} \quad \frac{d}{dt} \big( \mathcal{A}_{A_t} H_t \big) = i [H_t, \mathcal{A}_{A_t} H_t] + \mathcal{A}_{A_t} \dot{H}_t. \end{aligned}(Adjointness)[A,X],YHS=X,[A,Y]HS,(Bilinear)[H,A],[Y,A]HS=AAH,YHS,(Time-derivative)ddt(AAtHt)=i[Ht,AAtHt]+AAtH˙t.
The last identity follows from B t := ad A t B t := ad A t B_(t):=ad_(A_(t))B_t := \mathrm{ad}_{A_t}Bt:=adAt, B ˙ t = ad A ˙ t = i [ ad H t , B t ] B ˙ t = ad A ˙ t = i [ ad H t , B t ] B^(˙)_(t)=ad_(A^(˙)_(t))=i[ad_(H_(t)),B_(t)]\dot{B}_t = \mathrm{ad}_{\dot{A}_t} = i [\mathrm{ad}_{H_t}, B_t]B˙t=adA˙t=i[adHt,Bt] and ( B t 2 ) ˙ = i [ ad H t , B t 2 ] ( B t 2 ) ˙ = i [ ad H t , B t 2 ] (B_(t)^(2))^(˙)=i[ad_(H_(t)),B_(t)^(2)]\dot{(B_t^2)} = i [\mathrm{ad}_{H_t}, B_t^2](Bt2)˙=i[adHt,Bt2], hence d d t ( B t 2 H t ) = i [ H t , B t 2 H t ] + B t 2 H ˙ t d d t ( B t 2 H t ) = i [ H t , B t 2 H t ] + B t 2 H ˙ t (d)/(dt)(B_(t)^(2)H_(t))=i[H_(t),B_(t)^(2)H_(t)]+B_(t)^(2)H^(˙)_(t)\frac{d}{dt} (B_t^2 H_t) = i [H_t, B_t^2 H_t] + B_t^2 \dot{H}_tddt(Bt2Ht)=i[Ht,Bt2Ht]+Bt2H˙t.
Now compute
δ J = 0 T ( G ( A ) , K H S λ t h [ H , A ] , [ K ˙ + i [ K , H ] , A ] H S ) d t . δ J = 0 T ( G ( A ) , K H S λ t h [ H , A ] , [ K ˙ + i [ K , H ] , A ] H S ) d t . deltaJ=int_(0)^(T)((:G(A),K:)_(HS)-lambda_(th)(:[H,A],[K^(˙)+i[K,H],A]:)_(HS))dt.\delta \mathfrak{J} = \int_0^T \Big( \langle G(A), K \rangle_{\rm HS} - \lambda_{\rm th} \langle [H, A], [\dot{K} + i [K, H], A] \rangle_{\rm HS} \Big) \, dt.δJ=0T(G(A),KHSλth[H,A],[K˙+i[K,H],A]HS)dt.
Use the bilinear identity twice and integrate by parts in time (boundary vanishes by compact support of K K KKK):
δ J = 0 T ( G ( A ) , K H S + λ t h d d t ( A A H ) , K H S λ t h i A A H , [ K , H ] H S ) d t . δ J = 0 T ( G ( A ) , K H S + λ t h d d t ( A A H ) , K H S λ t h i A A H , [ K , H ] H S ) d t . deltaJ=int_(0)^(T)((:G(A),K:)_(HS)+lambda_(th)(:(d)/(dt)(A_(A)H),K:)_(HS)-lambda_(th)i(:A_(A)H,[K,H]:)_(HS))dt.\delta \mathfrak{J} = \int_0^T \Big( \langle G(A), K \rangle_{\rm HS} + \lambda_{\rm th} \big\langle \tfrac{d}{dt} (\mathcal{A}_A H), K \big\rangle_{\rm HS} - \lambda_{\rm th} i \langle \mathcal{A}_A H, [K, H] \rangle_{\rm HS} \Big) \, dt.δJ=0T(G(A),KHS+λthddt(AAH),KHSλthiAAH,[K,H]HS)dt.
Rewrite A A H , [ K , H ] H S = [ H , A A H ] , K H S A A H , [ K , H ] H S = [ H , A A H ] , K H S (:A_(A)H,[K,H]:)_(HS)=(:[H,A_(A)H],K:)_(HS)\langle \mathcal{A}_A H, [K, H] \rangle_{\rm HS} = \langle [H, \mathcal{A}_A H], K \rangle_{\rm HS}AAH,[K,H]HS=[H,AAH],KHS to obtain
δ J = 0 T G ( A ) + λ t h A A H ˙ , K H S d t . δ J = 0 T G ( A ) + λ t h A A H ˙ , K H S d t . deltaJ=int_(0)^(T)(:G(A)+lambda_(th)A_(A)H^(˙),K:)_(HS)dt.\delta \mathfrak{J} = \int_0^T \big\langle G(A) + \lambda_{\rm th} \mathcal{A}_A \dot{H}, K \big\rangle_{\rm HS} \, dt.δJ=0TG(A)+λthAAH˙,KHSdt.
Since K K KKK is arbitrary in C c 1 C c 1 C_(c)^(1)C_c^1Cc1, stationarity of J J J\mathfrak{J}J on unitary paths enforces A A H ˙ = ( 1 / λ t h ) G ( A ) A A H ˙ = ( 1 / λ t h ) G ( A ) A_(A)H^(˙)=-(1//lambda_(th))G(A)\mathcal{A}_A \dot{H} = -(1/\lambda_{\rm th}) G(A)AAH˙=(1/λth)G(A) on Ran A A Ran A A RanA_(A)\mathrm{Ran} \mathcal{A}_ARanAA.
This explicit computation shows why the earlier shortcut to G + λ t h A A H , K H S G + λ t h A A H , K H S (:G+lambda_(th)A_(A)H,K:)_(HS)\langle G + \lambda_{\rm th} \mathcal{A}_A H, K \rangle_{\rm HS}G+λthAAH,KHS is invalid and justifies, for dynamics, the pointwise/Pontryagin route used in the proof above.
Normalization and units.
Because the same normalized HS geometry defines (i) the predictive gradient and (ii) the orbit-throughput quadratic, the physical Planck constant is fixed by
= λ t h 1 . = λ t h 1 . ℏ=lambda_(th)^(-1).\boxed{\hbar = \lambda_{\rm th}^{-1}}.=λth1.
Calibration can be done operationally (e.g., two-level Fubini–Study speed) without ambiguity (§3.2.C).

3.2.C — Multiplier stability & metric uniqueness

Epi/Mosco stability across blocks.
With B o r b ; Λ ( A , H ) := 1 2 | [ H , A ] | H S ; Λ 2 B o r b ; Λ ( A , H ) := 1 2 | [ H , A ] | H S ; Λ 2 B_(orb;Lambda)(A,H):=(1)/(2)|[H,A]|_(HS;Lambda)^(2)\mathcal{B}_{{\rm orb};\Lambda}(A,H) := \tfrac{1}{2} |[H,A]|_{{\rm HS};\Lambda}^2Borb;Λ(A,H):=12|[H,A]|HS;Λ2 and B o r b := sup Λ B o r b ; Λ B o r b := sup Λ B o r b ; Λ B_(orb):=s u p _(Lambda)B_(orb;Lambda)\mathcal{B}_{\rm orb} := \sup_\Lambda \mathcal{B}_{{\rm orb};\Lambda}Borb:=supΛBorb;Λ, the value functions
Φ Λ := sup U 0 T ( S ( A t ) λ t h B o r b ; Λ ( A t , H t ) ) d t Φ Λ := sup U 0 T ( S ( A t ) λ t h B o r b ; Λ ( A t , H t ) ) d t Phi _(Lambda):=s u p_(U_(*))int_(0)^(T)(S(A_(t))-lambda_(th)B_(orb;Lambda)(A_(t),H_(t)))dt\Phi_\Lambda := \sup_{U_\cdot} \int_0^T \Big( \mathcal{S}(A_t) - \lambda_{\rm th} \mathcal{B}_{{\rm orb};\Lambda}(A_t, H_t) \Big) \, dtΦΛ:=supU0T(S(At)λthBorb;Λ(At,Ht))dt
epi-converge (Mosco) to Φ Φ Phi\PhiΦ defined with B o r b B o r b B_(orb)\mathcal{B}_{\rm orb}Borb.
At points of differentiability, multipliers converge: λ t h , Λ λ t h λ t h , Λ λ t h lambda_(th,Lambda)rarrlambda_(th)\lambda_{{\rm th},\Lambda} \to \lambda_{\rm th}λth,Λλth.
Hence = λ t h 1 = λ t h 1 ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1}=λth1 is independent of the exhausting sequence.
Ad-invariant metric uniqueness (up to scale).
On each finite block, any Ad-invariant inner product on observables is a scalar multiple α α alpha\alphaα of normalized HS (Schur).
Replacing , H S , H S (:*,*:)_(HS)\langle \cdot, \cdot \rangle_{\rm HS},HS by α , H S α , H S alpha(:*,*:)_(HS)\alpha \langle \cdot, \cdot \rangle_{\rm HS}α,HS rescales G α 1 G G α 1 G G|->alpha^(-1)GG \mapsto \alpha^{-1} GGα1G and | [ H , A ] | 2 α | [ H , A ] | 2 | [ H , A ] | 2 α | [ H , A ] | 2 |[H,A]|^(2)|->alpha|[H,A]|^(2)|[H,A]|^2 \mapsto \alpha |[H,A]|^2|[H,A]|2α|[H,A]|2.
The product α λ t h 1 α λ t h 1 alphalambda_(th)^(-1)\alpha \lambda_{\rm th}^{-1}αλth1 — i.e., \hbar — is invariant under this rescaling.
Thus \hbar is well-defined (after a single operational calibration).

3.3 — Hypotheses U (unbounded generators; GKSL on a core)

  • U1 (Core & closability). There exists a common invariant core D D D\mathcal{D}D for H H HHH and all L j L j L_(j)L_jLj; δ H δ H delta _(H)\delta_HδH is closable on A l o c A l o c A_(loc)\mathcal{A}_{\rm loc}Aloc.
  • U2 (Form bounds). There exists N 0 N 0 N >= 0N \ge 0N0 (number-type) and constants a < 1 a < 1 a < 1a < 1a<1, b < b < b < oob < \inftyb< with, for all ψ D ψ D psi inD\psi \in \mathcal{D}ψD,
H ψ 2 + j L j ψ 2 a N ψ 2 + b ψ 2 . H ψ 2 + j L j ψ 2 a N ψ 2 + b ψ 2 . ||H psi||^(2)+sum _(j)||L_(j)psi||^(2) <= a||N psi||^(2)+b||psi||^(2).\|H \psi\|^2 + \sum_j \|L_j \psi\|^2 \le a \|N \psi\|^2 + b \|\psi\|^2.Hψ2+jLjψ2aNψ2+bψ2.
  • U3 (Quasi-locality). Interactions have finite range and uniformly bounded overlap across Λ Λ Lambda\LambdaΛ.
  • U4 (Semigroup). The GKSL closure generates a unique strongly continuous CPTP semigroup with Lieb–Robinson-type bounds.
  • U5 (Lyapunov drift). There exist c 0 , c 1 > 0 c 0 , c 1 > 0 c_(0),c_(1) > 0c_0, c_1 > 0c0,c1>0 with L ( N ) c 0 c 1 N L ( N ) c 0 c 1 N L^(**)(N) <= c_(0)-c_(1)N\mathcal{L}^*(N) \le c_0 - c_1 NL(N)c0c1N on D D D\mathcal{D}D.
Drift ⇒ leakage finiteness.
For W = ( 1 + N ) s W = ( 1 + N ) s W=(1+N)^(-s)W = (1+N)^{-s}W=(1+N)s with s > 1 / 2 s > 1 / 2 s > 1//2s > 1/2s>1/2,
j ω ( L j W L j ) < B l e a k ( { L j } , W ) < , j ω ( L j W L j ) < B l e a k ( { L j } , W ) < , sum _(j)omega(L_(j)^(†)WL_(j)) < ooquad=>quadB_(leak)({L_(j)},W) < oo,\sum_j \omega(L_j^\dagger W L_j) < \infty \quad \Rightarrow \quad \mathcal{B}_{\rm leak}(\{L_j\}, W) < \infty,jω(LjWLj)<Bleak({Lj},W)<,
by Cauchy–Schwarz in the ω ω omega\omegaω-HS norm and U5.

3.4 — Zero-leakage ⇒ unitary GKSL

Let L = i [ H , ] + j L j ( ) L j 1 2 { L j L j , } L = i [ H , ] + j L j ( ) L j 1 2 { L j L j , } L=i[H,*]+sum _(j)L_(j)^(†)(*)L_(j)-(1)/(2){L_(j)^(†)L_(j),*}\mathcal{L} = i [H, \cdot] + \sum_j L_j^\dagger (\cdot) L_j - \tfrac{1}{2} \{L_j^\dagger L_j, \cdot\}L=i[H,]+jLj()Lj12{LjLj,} satisfy U1–U4, and let W 0 W 0 W>-0W \succ 0W0.
Proposition 3.4.1. If B l e a k ( { L j } , W ) = j ω ( L j W L j ) = 0 B l e a k ( { L j } , W ) = j ω ( L j W L j ) = 0 B_(leak)({L_(j)},W)=sum _(j)omega(L_(j)^(†)WL_(j))=0\mathcal{B}_{\rm leak}(\{L_j\}, W) = \sum_j \omega(L_j^\dagger W L_j) = 0Bleak({Lj},W)=jω(LjWLj)=0, then L j = 0 L j = 0 L_(j)=0L_j = 0Lj=0 for all j j jjj and the semigroup is unitary with generator H H HHH on the core.
Proof. Each term ω ( L j W L j ) = | W 1 / 2 L j | 2 , ω 2 0 ω ( L j W L j ) = | W 1 / 2 L j | 2 , ω 2 0 omega(L_(j)^(†)WL_(j))=|W^(1//2)L_(j)|_(2,omega)^(2) >= 0\omega(L_j^\dagger W L_j) = |W^{1/2} L_j|_{2,\omega}^2 \ge 0ω(LjWLj)=|W1/2Lj|2,ω20. If the sum vanishes, W 1 / 2 L j = 0 W 1 / 2 L j = 0 W^(1//2)L_(j)=0W^{1/2} L_j = 0W1/2Lj=0; since W 0 W 0 W>-0W \succ 0W0, L j = 0 L j = 0 L_(j)=0L_j = 0Lj=0. \square
Thus, in the zero-leakage limit, the fast sector is purely unitary, governed by H H HHH from §3.2.

3.5 — Pointer alignment (unitary-orbit minimizer)

Fix W 0 W 0 W>-0W \succ 0W0. Consider a noise block with fixed singular values (fixed “strength spectrum”). For C C CCC in this block define
L ( C ) := Tr ( W 1 / 2 C C W 1 / 2 ) . L ( C ) := Tr ( W 1 / 2 C C W 1 / 2 ) . L(C):=Tr(W^(1//2)C^(†)CW^(1//2)).\mathcal{L}(C) := \operatorname{Tr} \big( W^{1/2} C^\dagger C \, W^{1/2} \big).L(C):=Tr(W1/2CCW1/2).
Theorem 3.5.1 (unitary-orbit rearrangement). Over the orbit { U C V : U , V unitary } { U C V : U , V  unitary } {UCV:U,V" unitary"}\{U C V : U, V \text{ unitary}\}{UCV:U,V unitary},
L ( U C V ) is minimized iff [ C C , W ] = 0. L ( U C V )  is minimized iff  [ C C , W ] = 0. L(UCV)" is minimized iff "[C^(†)C,W]=0.\boxed{\mathcal{L}(U C V) \text{ is minimized iff } [C^\dagger C, W] = 0.}L(UCV) is minimized iff [CC,W]=0.
Equivalently, minimizers co-diagonalize C C C C C^(†)CC^\dagger CCC and W W WWW, selecting the pointer basis (degeneracies handled blockwise).
Proof. L ( U C V ) = Tr ( W 1 / 2 V C C V W 1 / 2 ) L ( U C V ) = Tr ( W 1 / 2 V C C V W 1 / 2 ) L(UCV)=Tr(W^(1//2)V^(†)C^(†)CVW^(1//2))\mathcal{L}(U C V) = \operatorname{Tr}(W^{1/2} V^\dagger C^\dagger C V W^{1/2})L(UCV)=Tr(W1/2VCCVW1/2). Let W = k w k Q k W = k w k Q k W=sum _(k)w_(k)Q_(k)W = \sum_k w_k Q_kW=kwkQk with w 1 w d > 0 w 1 w d > 0 w_(1) >= cdots >= w_(d) > 0w_1 \ge \cdots \ge w_d > 0w1wd>0. By von Neumann/Schur–Horn, with fixed eigenvalues { σ } { σ } {sigma_(ℓ)}\{\sigma_\ell\}{σ} of C C C C C^(†)CC^\dagger CCC, the minimum of Tr ( W , V C C V ) Tr ( W , V C C V ) Tr(W,V^(†)C^(†)CV)\operatorname{Tr}(W, V^\dagger C^\dagger C V)Tr(W,VCCV) is attained when V V VVV aligns eigenvectors so that C C C C C^(†)CC^\dagger CCC is diagonal in the eigenbasis of W W WWW. Equality enforces [ C C , W ] = 0 [ C C , W ] = 0 [C^(†)C,W]=0[C^\dagger C, W] = 0[CC,W]=0. \square

3.6 — Summary

  • Pricing the orbit speed $|[H,A]|_{\rm HS}$ (rather than the Ad-invariant superoperator norm) and matching the same normalized HS geometry for gradient and quadratic yields
    [
    \dot A=\frac{i}{\hbar}[H^*(A),A],\qquad
    H^*(A)=\mathcal A_A^{+}!\left(\frac{-,i}{\lambda_{\rm th}}[A,G(A)]\right),\quad \hbar=\lambda_{\rm th}^{-1}.
    ]
    Here $\mathcal A_A=[A,[A,\cdot]]\ge0$ and $\mathcal A_A^+$ projects off the commutant.
  • Stability. Epi/Mosco limits across blocks preserve $\lambda_{\rm th}$; Ad-invariant metric rescalings cancel in $\hbar$. A single operational calibration fixes $\hbar$.
  • Fast sector. Zero leakage forces GKSL to reduce to a unitary group. For nonzero leakage, pointer alignment minimizes $\operatorname{Tr}(W{1/2}C\dagger C W^{1/2})$ by co-diagonalizing $C^\dagger C$ with $W$.
This completes the fast-sector normalization and resolves the stationarity/dynamics gap with explicit variational calculus and a pointwise/Pontryagin control derivation in a C*-compatible, mathematically airtight manner.

Chapter 4 — Scaffold Selection: Γ‑Limit ⇒ Einstein–Hilbert (+ compactness)

Scope. We state a clean cone‑preserving topology after gauge‑fixing and prove equi‑coercivity and Γ‑compactness. This slots into the EH Γ‑limit derivation and enforces single‑cone microhyperbolicity.

4.1 Γ‑compactness in a cone‑preserving class (Theorem 4.1′)

Setup. Bounded‑geometry manifold: uniform injectivity radius and bounded curvature (all derivatives). Fix de Donder gauge inside the single‑cone class; use harmonic coordinates on charts.
Cone-preserving topology. g H l o c 2 g H l o c 2 g inH_(loc)^(2)g \in H^2_{\rm loc}gHloc2 and for reference metrics g , g g , g g_(**),g^(**)g_*, g^*g,g on compacts,
g g g a.e. in harmonic coordinates (cone inclusion). g g g a.e. in harmonic coordinates (cone inclusion). g_(**) <= g <= g^(**)quad"a.e. in harmonic coordinates (cone inclusion)."g_* \le g \le g^* \quad \text{a.e. in harmonic coordinates (cone inclusion).}ggga.e. in harmonic coordinates (cone inclusion).
Uniform symbol bounds. There exist 0 < λ Λ 0 < λ Λ 0 < lambda <= Lambda0<\lambda\le\Lambda0<λΛ such that for all admissible g g ggg and ξ ξ xi\xiξ,
λ | ξ | 4 σ ( A ε ( g ) ) [ ξ ] Λ | ξ | 4 (parameter‑ellipticity on compacts). λ | ξ | 4 σ ( A ε ( g ) ) [ ξ ] Λ | ξ | 4 (parameter‑ellipticity on compacts). lambda|xi|^(4) <= sigma(A_(epsi)(g))[xi] <= Lambda|xi|^(4)qquad(parameter‑ellipticity on compacts).\lambda\,|\xi|^4\ \le\ \sigma(\mathbb A_\varepsilon(g))[\xi]\ \le\ \Lambda\,|\xi|^4\qquad\text{(parameter‑ellipticity on compacts).}λ|ξ|4  σ(Aε(g))[ξ]  Λ|ξ|4(parameter‑ellipticity on compacts).
Gårding inequality (uniform). There exist c 1 , c 2 , c 3 > 0 c 1 , c 2 , c 3 > 0 c_(1),c_(2),c_(3) > 0c_1,c_2,c_3>0c1,c2,c3>0, independent of ε ε epsi\varepsilonε and g g ggg, with
F ε ( g ) c 1 2 g L 2 2 c 2 g H 1 2 c 3 . F ε ( g ) c 1 2 g L 2 2 c 2 g H 1 2 c 3 . F_(epsi)(g) >= c_(1)||grad^(2)g||_(L^(2))^(2)-c_(2)||g||_(H^(1))^(2)-c_(3).\boxed{\ \mathcal F_\varepsilon(g)\ \ge\ c_1\|\nabla^2 g\|_{L^2}^2\ -\ c_2\|g\|_{H^1}^2\ -\ c_3\ }. Fε(g)  c12gL22  c2gH12  c3 .
Boundary/at‑infinity control. If M M MMM is noncompact, either (i) impose uniform equivalence to a reference g g g_( oo)g_\inftyg outside a compact set, or (ii) include in V ε V ε V_( epsi)V_\varepsilonVε a cutoff penalizing deviations at infinity to control g H 1 g H 1 ||g||_(H^(1))\|g\|_{H^1}gH1.

Theorem 4.1′ (equi‑coercivity and Γ‑limit)

The family { F ε } { F ε } {F_(epsi)}\{\mathcal F_\varepsilon\}{Fε} is equi‑coercive in weak H 2 H 2 H^(2)H^2H2; any weak H 2 H 2 H^(2)H^2H2 cluster point of minimizers is a minimizer of F 0 F 0 F_(0)\mathcal F_0F0 (Γ‑liminf and recovery hold).
Proof sketch. Gauge‑fixing removes diffeomorphism degeneracy; bounded geometry yields Rellich compactness in weak H 2 H 2 H^(2)H^2H2; single‑cone stability controls the principal symbol. Γ‑compactness follows from equi‑coercivity and lower semicontinuity; recoveries are built by mollification in harmonic charts and partition‑of‑unity gluing.

Chapter 5 — Coupled Laws (Einstein–YM + GKSL via Joint Stationarity)

Scope. We prove the coupled slow–fast laws at joint stationarity of the coherence program. The argument rests on:
  1. a directional envelope theorem for worst-case pokes with direction-selecting minimizers (no illicit inf–variation swap);
  2. effective sources T e f f , J e f f T e f f , J e f f T^(eff),J^(eff)T^{\rm eff},J^{\rm eff}Teff,Jeff defined first in the Clarke framework (existence + conservation without linearity), then upgraded to unique tensors under an explicit Gateaux/Clarke condition;
  3. first-variation convergence for the slow Γ-limit via localized Mosco/Attouch;
  4. a fast-sector control law that prices the derivation [ H , ] [ H , ] [H,*][H,\cdot][H,] (the actual throughput), uses the Moore–Penrose pseudoinverse of the orbit Laplacian A A = ad A ad A A A = ad A ad A A_(A)=ad_(A)^(**)ad_(A)\mathcal A_A=\mathrm{ad}_A^{\,*}\mathrm{ad}_AAA=adAadA, and yields the Heisenberg law with a consistent normalization = λ t h 1 = λ t h 1 ℏ=lambda_(th)^(-1)\hbar=\lambda_{\rm th}^{-1}=λth1.

5.1 — Admissible variables and objective

Slow variables. Lorentzian metrics g g ggg of bounded geometry in the single-cone class (weak H l o c 2 H l o c 2 H_(loc)^(2)H^2_{\rm loc}Hloc2, de Donder gauge on compacta) and compact-group connections A A AAA with curvature F L l o c 2 F L l o c 2 F inL_(loc)^(2)F\in L^2_{\rm loc}FLloc2.
Fast variables. A GKSL generator on the quasi-local C C C^(**)C^*C-algebra (Ch. 3):
L ( ρ ) = i [ H , ρ ] + j L j ρ L j 1 2 { L j L j , ρ } , L ( ρ ) = i [ H , ρ ] + j L j ρ L j 1 2 { L j L j , ρ } , L(rho)=-i[H,rho]+sum _(j)L_(j)rhoL_(j)^(†)-(1)/(2){L_(j)^(†)L_(j),rho},\mathcal{L}(\rho) = -i [H, \rho] + \sum_j L_j \rho L_j^\dagger - \tfrac{1}{2} \{ L_j^\dagger L_j, \rho \},L(ρ)=i[H,ρ]+jLjρLj12{LjLj,ρ},
with budgets B t h , B c x , B l e a k B t h , B c x , B l e a k B_(th),B_(cx),B_(leak)\mathcal{B}_{\rm th}, \mathcal{B}_{\rm cx}, \mathcal{B}_{\rm leak}Bth,Bcx,Bleak as in Appendix A and Hypotheses U.
Pokes. P P ¯ bar(P)\overline{\mathcal P}P: diamond-norm closure of the causal, Γ-local cone (§2.2).
Slow action (Γ-limit). Assumption D ^(***)^\star: a family { F ε ( g , A ) } { F ε ( g , A ) } {F_(epsi)(g,A)}\{\mathcal F_\varepsilon(g,A)\}{Fε(g,A)} Γ-converges on the cone-class to
F 0 ( g , A ) = 1 16 π G M ( R 2 Λ ) | g | d d x + 1 2 g Y M 2 M tr ( F F ) , F 0 ( g , A ) = 1 16 π G M ( R 2 Λ ) | g | d d x + 1 2 g Y M 2 M tr ( F F ) , F_(0)(g,A)=(1)/(16 pi G)int _(M)(R-2Lambda)sqrt(|g|)d^(d)x+(1)/(2g_(YM)^(2))int _(M)tr(F^^**F),\mathcal F_0(g,A)=\frac{1}{16\pi G}\!\int_M (R-2\Lambda)\sqrt{|g|}\,d^dx\;+\;\frac{1}{2g_{\rm YM}^2}\!\int_M \mathrm{tr}(F\wedge\!*F),F0(g,A)=116πGM(R2Λ)|g|ddx+12gYM2Mtr(FF),
with equi-coercivity and boundary control on compacta.
Objective.
J ( g , A , H , { L j } ) := V ( g , A ) λ t h B t h ( H ) λ c x B c x ( L ) λ l e a k B l e a k ( { L j } ) F ε ( g , A ) J ( g , A , H , { L j } ) := V ( g , A ) λ t h B t h ( H ) λ c x B c x ( L ) λ l e a k B l e a k ( { L j } ) F ε ( g , A ) J(g,A,H,{L_(j)}):=V(g,A)-lambda_(th)B_(th)(H)-lambda_(cx)B_(cx)(L)-lambda_(leak)B_(leak)({L_(j)})-F_(epsi)(g,A)\boxed{ \mathcal J(g,A,H,\{L_j\})\ :=\ V(g,A)\ -\ \lambda_{\rm th}\mathcal B_{\rm th}(H)\ -\ \lambda_{\rm cx}\mathcal B_{\rm cx}(\mathcal L)\ -\ \lambda_{\rm leak}\mathcal B_{\rm leak}(\{L_j\})\ -\ \mathcal F_\varepsilon(g,A) }J(g,A,H,{Lj}) := V(g,A)  λthBth(H)  λcxBcx(L)  λleakBleak({Lj})  Fε(g,A)
where V ( g , A ) := inf Φ P CL ( g , A ; Φ ) V ( g , A ) := inf Φ P ¯ CL ( g , A ; Φ ) V(g,A):=i n f_(Phi in bar(P))CL(g,A;Phi)V(g,A):=\inf_{\Phi\in\overline{\mathcal P}}\mathrm{CL}(g,A;\Phi)V(g,A):=infΦPCL(g,A;Φ). Slater interior holds (Appendix E).

5.2 — Parametric envelope for worst-case pokes (directional form)

We work with directional derivatives and avoid over-claiming Gateaux differentiability.

Hypotheses E′ (attainment, regularity, direction-wise selection)

  • E1 (attainment/compactness). For each ( g , A ) ( g , A ) (g,A)(g,A)(g,A), Φ CL ( g , A ; Φ ) Φ CL ( g , A ; Φ ) Phi|->CL(g,A;Phi)\Phi\mapsto \mathrm{CL}(g,A;\Phi)ΦCL(g,A;Φ) is l.s.c. and inf-compact on P P ¯ bar(P)\overline{\mathcal P}P; hence
    M ( g , A ) := Argmin Φ P CL ( g , A ; Φ ) M ( g , A ) := Argmin Φ P ¯ CL ( g , A ; Φ ) M(g,A):=Argmin_(Phi in bar(P))CL(g,A;Phi)\mathsf M(g,A):=\operatorname{Argmin}_{\Phi\in\overline{\mathcal P}}\mathrm{CL}(g,A;\Phi)M(g,A):=ArgminΦPCL(g,A;Φ)
    is non-empty and compact.
  • E2 (Carathéodory). ( g , A , Φ ) CL ( g , A ; Φ ) ( g , A , Φ ) CL ( g , A ; Φ ) (g,A,Phi)|->CL(g,A;Phi)(g,A,\Phi)\mapsto \mathrm{CL}(g,A;\Phi)(g,A,Φ)CL(g,A;Φ) is continuous in ( g , A ) ( g , A ) (g,A)(g,A)(g,A) for fixed Φ Φ Phi\PhiΦ, and l.s.c. in Φ Φ Phi\PhiΦ for fixed ( g , A ) ( g , A ) (g,A)(g,A)(g,A).
  • E3 (directional differentiability). For each Φ Φ Phi\PhiΦ and cone-admissible variation δ = ( δ g , δ A ) δ = ( δ g , δ A ) delta=(delta g,delta A)\delta=(\delta g,\delta A)δ=(δg,δA) with compact support,
    D CL ( g , A ; Φ ; δ ) := lim t 0 CL ( g + t δ g , A + t δ A ; Φ ) CL ( g , A ; Φ ) t D CL ( g , A ; Φ ; δ ) := lim t 0 CL ( g + t δ g , A + t δ A ; Φ ) CL ( g , A ; Φ ) t DCL(g,A;Phi;delta):=lim_(t darr0)(CL(g+t delta g,A+t delta A;Phi)-CL(g,A;Phi))/(t)D\,\mathrm{CL}(g,A;\Phi;\delta):=\lim_{t\downarrow0}\frac{\mathrm{CL}(g+t\delta g,A+t\delta A;\Phi)-\mathrm{CL}(g,A;\Phi)}{t}DCL(g,A;Φ;δ):=limt0CL(g+tδg,A+tδA;Φ)CL(g,A;Φ)t
    exists and is positively homogeneous in δ δ delta\deltaδ.
  • E4 (equi-differentiability on argmin graphs). There exist M > 0 M > 0 M > 0M>0M>0 and a neighborhood U ( g , A ) U ( g , A ) U∋(g,A)U\ni(g,A)U(g,A) such that
    | D CL ( g , A ; Φ ; δ ) | M δ | D CL ( g , A ; Φ ; δ ) | M δ |DCL(g^('),A^(');Phi;delta)| <= M||delta|||D\,\mathrm{CL}(g',A';\Phi;\delta)|\le M\|\delta\||DCL(g,A;Φ;δ)|Mδ
    for all ( g , A ) U ( g , A ) U (g^('),A^('))in U(g',A')\in U(g,A)U, Φ M ( g , A ) Φ M ( g , A ) Phi inM(g^('),A^('))\Phi\in\mathsf M(g',A')ΦM(g,A), and admissible δ δ delta\deltaδ.
  • E5 (direction-wise selection). For each ( g , A ) ( g , A ) (g,A)(g,A)(g,A) and direction δ δ delta\deltaδ, choose
    Φ ( g , A ; δ ) arg min Φ M ( g , A ) D CL ( g , A ; Φ ; δ ) Φ ( g , A ; δ ) arg min Φ M ( g , A ) D CL ( g , A ; Φ ; δ ) Phi^(**)(g,A;delta)in arg min_(Phi inM(g,A))DCL(g,A;Phi;delta)\Phi^*(g,A;\delta)\in\arg\min_{\Phi\in\mathsf M(g,A)} D\,\mathrm{CL}(g,A;\Phi;\delta)Φ(g,A;δ)argminΦM(g,A)DCL(g,A;Φ;δ)
    (measurable selection exists by Kuratowski–Ryll-Nardzewski).

Theorem 5.2′ (Directional envelope)

Under E1–E5, for every cone-admissible δ δ delta\deltaδ,
D V ( g , A ; δ ) = min Φ M ( g , A ) D CL ( g , A ; Φ ; δ ) = D CL ( g , A ; Φ ( g , A ; δ ) ; δ ) . D V ( g , A ; δ ) = min Φ M ( g , A ) D CL ( g , A ; Φ ; δ ) = D CL ( g , A ; Φ ( g , A ; δ ) ; δ ) . DV(g,A;delta)=min_(Phi inM(g,A))DCL(g,A;Phi;delta)=DCL(g,A;Phi^(**)(g,A;delta);delta).\boxed{\ D V(g,A;\delta)\ =\ \min_{\Phi\in\mathsf M(g,A)} D\,\mathrm{CL}(g,A;\Phi;\delta)\ =\ D\,\mathrm{CL}\big(g,A;\Phi^*(g,A;\delta);\delta\big). \ } DV(g,A;δ) = minΦM(g,A)DCL(g,A;Φ;δ) = DCL(g,A;Φ(g,A;δ);δ). 

Corollary 5.2b (Gateaux differentiability — two routes)

At ( g , A ) ( g , A ) (g,A)(g,A)(g,A), V V VVV is Gateaux differentiable (two-sided, linear in δ δ delta\deltaδ) if either:
  1. (Unique active). A strictly convex, cone-local tie-breaker (vanishing in the Γ-limit) makes M ( g , A ) M ( g , A ) M(g,A)\mathsf M(g,A)M(g,A) a singleton with locally Lipschitz dependence; then D V ( g , A ; δ ) = D CL ( g , A ; Φ ( g , A ) ; δ ) D V ( g , A ; δ ) = D CL ( g , A ; Φ ( g , A ) ; δ ) DV(g,A;delta)=DCL(g,A;Phi^(**)(g,A);delta)D V(g,A;\delta)=D\,\mathrm{CL}(g,A;\Phi^*(g,A);\delta)DV(g,A;δ)=DCL(g,A;Φ(g,A);δ).
  2. (Clarke-regular active set). M ( g , A ) M ( g , A ) M(g,A)\mathsf M(g,A)M(g,A) is compact and D CL ( g , A ; Φ ; δ ) D CL ( g , A ; Φ ; δ ) DCL(g,A;Phi;delta)D\,\mathrm{CL}(g,A;\Phi;\delta)DCL(g,A;Φ;δ) is constant over Φ M ( g , A ) Φ M ( g , A ) Phi inM(g,A)\Phi\in\mathsf M(g,A)ΦM(g,A) for each δ δ delta\deltaδ. Then right/left derivatives coincide and are linear.

5.3 — Effective stress tensor and current (Clarke framework → upgrade)

We first ensure existence and conservation of sources without assuming linearity (Clarke framework), then upgrade to unique tensors under Cor. 5.2b.

Assumption L0 (local Lipschitz of the envelope)

Under E1–E4, the value function V ( g , A ) = min Φ M ( g , A ) CL ( g , A ; Φ ) V ( g , A ) = min Φ M ( g , A ) CL ( g , A ; Φ ) V(g,A)=min_(Phi inM(g,A))CL(g,A;Phi)V(g,A)=\min_{\Phi\in\mathsf M(g,A)}\mathrm{CL}(g,A;\Phi)V(g,A)=minΦM(g,A)CL(g,A;Φ) is locally Lipschitz on the cone-class (Danskin–Rockafellar with the uniform bound in E4).
Localization for Clarke calculus. All Clarke sub/super-differential objects are taken on bounded subdomains Ω M Ω M Omega⋐M\Omega\Subset MΩM with compactly supported variations; conclusions pass to M M MMM by exhaustion Ω n M Ω n M Omega _(n)uarr M\Omega_n\uparrow MΩnM and compatibility of the cone-class cutoffs. This keeps the Banach-space hypotheses crisp and avoids overreach beyond bounded domains.
Clarke subdifferentials. Let g C V ( g , A ) H l o c 2 g C V ( g , A ) H l o c 2 del_(g)^(C)V(g,A)subH_(loc)^(-2)\partial_g^{\mathrm C} V(g,A)\subset H^{-2}_{\rm loc}gCV(g,A)Hloc2 and A C V ( g , A ) H l o c 1 A C V ( g , A ) H l o c 1 del_(A)^(C)V(g,A)subH_(loc)^(-1)\partial_A^{\mathrm C} V(g,A)\subset H^{-1}_{\rm loc}ACV(g,A)Hloc1 denote the Clarke subdifferentials; V g ( g , A ; δ g ) V g ( g , A ; δ g ) V_(g)^(@)(g,A;delta g)V_g^\circ(g,A;\delta g)Vg(g,A;δg), V A ( g , A ; δ A ) V A ( g , A ; δ A ) V_(A)^(@)(g,A;delta A)V_A^\circ(g,A;\delta A)VA(g,A;δA) are the Clarke directional derivatives.

5.3.1 Subgradient sources (always well-posed) and conservation

Take any T e f f g C V ( g , A ) T e f f g C V ( g , A ) T^(eff)indel_(g)^(C)V(g,A)T^{\rm eff}\in\partial_g^{\mathrm C}V(g,A)TeffgCV(g,A), J e f f A C V ( g , A ) J e f f A C V ( g , A ) J^(eff)indel_(A)^(C)V(g,A)J^{\rm eff}\in\partial_A^{\mathrm C}V(g,A)JeffACV(g,A) so that
T e f f , δ g V g ( g , A ; δ g ) , J e f f , δ A V A ( g , A ; δ A ) . T e f f , δ g V g ( g , A ; δ g ) , J e f f , δ A V A ( g , A ; δ A ) . (:T^(eff),delta g:) <= V_(g)^(@)(g,A;delta g),qquad(:J^(eff),delta A:) <= V_(A)^(@)(g,A;delta A).\langle T^{\rm eff},\delta g\rangle\ \le\ V_g^\circ(g,A;\delta g),\qquad \langle J^{\rm eff},\delta A\rangle\ \le\ V_A^\circ(g,A;\delta A).Teff,δg  Vg(g,A;δg),Jeff,δA  VA(g,A;δA).
If CL CL CL\mathrm{CL}CL is diffeomorphism- and gauge-covariant on the cone-class, then for compactly supported X , χ X , χ X,chiX,\chiX,χ,
V g ( g , A ; L X g ) = V g ( g , A ; L X g ) = 0 , V A ( g , A ; D χ ) = V A ( g , A ; D χ ) = 0 , V g ( g , A ; L X g ) = V g ( g , A ; L X g ) = 0 , V A ( g , A ; D χ ) = V A ( g , A ; D χ ) = 0 , V_(g)^(@)(g,A;L_(X)g)=V_(g)^(@)(g,A;-L_(X)g)=0,qquadV_(A)^(@)(g,A;D chi)=V_(A)^(@)(g,A;-D chi)=0,V_g^\circ(g,A;\mathcal L_X g)=V_g^\circ(g,A;-\mathcal L_X g)=0,\qquad V_A^\circ(g,A;D\chi)=V_A^\circ(g,A;-D\chi)=0,Vg(g,A;LXg)=Vg(g,A;LXg)=0,VA(g,A;Dχ)=VA(g,A;Dχ)=0,
hence
T e f f , L X g = 0 , J e f f , D χ = 0. T e f f , L X g = 0 , J e f f , D χ = 0. (:T^(eff),L_(X)g:)=0,qquad(:J^(eff),D chi:)=0.\langle T^{\rm eff},\mathcal L_X g\rangle=0,\qquad \langle J^{\rm eff},D\chi\rangle=0.Teff,LXg=0,Jeff,Dχ=0.
Integrating by parts within the cone-class yields the weak conservation laws
μ T μ ν e f f = 0 , D μ J μ e f f = 0 μ T μ ν e f f = 0 , D μ J μ e f f = 0 grad ^(mu)T_(mu nu)^(eff)=0,qquadD^( mu)J_( mu)^(eff)=0\nabla^\mu T^{\rm eff}_{\mu\nu}=0,\qquad D^\mu J^{\rm eff}_\mu=0μTμνeff=0,DμJμeff=0
for every Clarke-selection T e f f , J e f f T e f f , J e f f T^(eff),J^(eff)T^{\rm eff},J^{\rm eff}Teff,Jeff.

5.3.2 Upgrade to single tensors (Distributional representation (Riesz on bounded domains); symmetry)

Assumption L (linearity at ( g , A ) ( g , A ) (g,A)(g,A)(g,A)). One of Cor. 5.2b’s conditions holds at ( g , A ) ( g , A ) (g,A)(g,A)(g,A). Then V V VVV is Gateaux differentiable in g , A g , A g,Ag,Ag,A; the Clarke subdifferentials are singletons and coincide with the Fréchet derivatives.
Corollary. Under Assumption L there exist unique distributions T e f f H l o c 2 T e f f H l o c 2 T^(eff)inH_(loc)^(-2)T^{\rm eff}\in H^{-2}_{\rm loc}TeffHloc2, J e f f H l o c 1 J e f f H l o c 1 J^(eff)inH_(loc)^(-1)J^{\rm eff}\in H^{-1}_{\rm loc}JeffHloc1 such that on each bounded domain Ω M Ω M Omega⋐M\Omega\Subset MΩM
D g V ( g , A ) [ δ g ] = 1 2 Ω T μ ν e f f δ g μ ν | g | d d x , D A V ( g , A ) [ δ A ] = Ω J μ e f f , δ A μ | g | d d x , D g V ( g , A ) [ δ g ] = 1 2 Ω T μ ν e f f δ g μ ν | g | d d x , D A V ( g , A ) [ δ A ] = Ω J μ e f f , δ A μ | g | d d x , {:[D_(g)V(g","A)[delta g]=(1)/(2)int _(Omega)T_(mu nu)^(eff)deltag^(mu nu)sqrt(|g|)d^(d)x","],[D_(A)V(g","A)[delta A]=int _(Omega)(:J_( mu)^(eff)","deltaA^( mu):)sqrt(|g|)d^(d)x","]:}\begin{aligned} D_g V(g,A)[\delta g] &= \tfrac12\int_\Omega T^{\rm eff}_{\mu\nu}\,\delta g^{\mu\nu}\,\sqrt{|g|}\,d^dx,\\ D_A V(g,A)[\delta A] &= \int_\Omega \langle J^{\rm eff}_\mu,\,\delta A^\mu\rangle\,\sqrt{|g|}\,d^dx, \end{aligned}DgV(g,A)[δg]=12ΩTμνeffδgμν|g|ddx,DAV(g,A)[δA]=ΩJμeff,δAμ|g|ddx,
with T μ ν e f f = T ν μ e f f T μ ν e f f = T ν μ e f f T_(mu nu)^(eff)=T_(nu mu)^(eff)T^{\rm eff}_{\mu\nu}=T^{\rm eff}_{\nu\mu}Tμνeff=Tνμeff; the representations are compatible under exhaustion and define T e f f , J e f f T e f f , J e f f T^(eff),J^(eff)T^{\rm eff},J^{\rm eff}Teff,Jeff globally on M M MMM. Conservation from §5.3.1 persists.

5.4 — Slow-sector Euler–Lagrange: EH + YM (first-variation convergence)

Γ-convergence alone does not yield convergence of first variations. We adopt a localized Mosco/Attouch scheme.
Assumption M (localized Mosco/Attouch). On each bounded Ω M Ω M Omega⋐M\Omega\Subset MΩM, F ε | Ω F ε | Ω F_(epsi)|_(Omega)\mathcal F_\varepsilon|_\OmegaFε|Ω are equi-coercive, l.s.c., and admit integral representations with Carathéodory integrands f ε ( x , ) f ε ( x , ) f_( epsi)(x,*)f_\varepsilon(x,\cdot)fε(x,) obeying uniform growth/ellipticity and f ε f 0 f ε f 0 f_( epsi)rarrf_(0)f_\varepsilon\to f_0fεf0 in L ( Ω ) L ( Ω ) L^( oo)(Omega)L^\infty(\Omega)L(Ω). Then F ε | Ω g r a p h F 0 | Ω F ε | Ω g r a p h F 0 | Ω delF_(epsi)|_(Omega)rarr"graph"delF_(0)|_(Omega)\partial \mathcal F_\varepsilon|_\Omega \xrightarrow{\rm graph} \partial \mathcal F_0|_\OmegaFε|ΩgraphF0|Ω (Attouch). Exhaust Ω n M Ω n M Omega _(n)uarr M\Omega_n\uparrow MΩnM.
Lemma 5.4.1 (convergence of first variations). For any admissible ( g , A ) ( g , A ) (g,A)(g,A)(g,A) and compactly supported cone-preserving ( δ g , δ A ) ( δ g , δ A ) (delta g,delta A)(\delta g,\delta A)(δg,δA),
lim ε 0 D F ε ( g , A ) [ δ g , δ A ] = D F 0 ( g , A ) [ δ g , δ A ] , lim ε 0 D F ε ( g , A ) [ δ g , δ A ] = D F 0 ( g , A ) [ δ g , δ A ] , lim_(epsi rarr0)DF_(epsi)(g,A)[delta g,delta A]=DF_(0)(g,A)[delta g,delta A],\lim_{\varepsilon\to0} D\,\mathcal F_\varepsilon(g,A)[\delta g,\delta A]\ =\ D\,\mathcal F_0(g,A)[\delta g,\delta A],limε0DFε(g,A)[δg,δA] = DF0(g,A)[δg,δA],
with
D g F 0 ( g , A ) [ δ g ] = 1 2 1 8 π G ( G μ ν + Λ g μ ν ) δ g μ ν | g | , D A F 0 ( g , A ) [ δ A ] = D α F α μ , δ A μ | g | . D g F 0 ( g , A ) [ δ g ] = 1 2 1 8 π G ( G μ ν + Λ g μ ν ) δ g μ ν | g | , D A F 0 ( g , A ) [ δ A ] = D α F α μ , δ A μ | g | . D_(g)F_(0)(g,A)[delta g]=(1)/(2)int(1)/(8pi G)(G_(mu nu)+Lambdag_(mu nu))deltag^(mu nu)sqrt(|g|),quadD_(A)F_(0)(g,A)[delta A]=int(:D^( alpha)F_(alpha mu),deltaA^( mu):)sqrt(|g|).D_g \mathcal F_0(g,A)[\delta g]=\tfrac12\!\int \!\tfrac{1}{8\pi G}\big(G_{\mu\nu}+\Lambda g_{\mu\nu}\big)\,\delta g^{\mu\nu}\sqrt{|g|},\quad D_A \mathcal F_0(g,A)[\delta A]=\int\!\langle D^\alpha F_{\alpha\mu},\delta A^\mu\rangle\sqrt{|g|}.DgF0(g,A)[δg]=1218πG(Gμν+Λgμν)δgμν|g|,DAF0(g,A)[δA]=DαFαμ,δAμ|g|.
Theorem 5.4.2 (Einstein–YM with operational sources).
Let ( g , A , H , { L j } ) ( g , A , H , { L j } ) (g,A,H,{L_(j)})(g,A,H,\{L_j\})(g,A,H,{Lj}) be a cone-class KKT point of J J J\mathcal JJ and assume Assumption L at ( g , A ) ( g , A ) (g,A)(g,A)(g,A). Then
G μ ν + Λ g μ ν = 8 π G T μ ν e f f , D α F α β = J β e f f , G μ ν + Λ g μ ν = 8 π G T μ ν e f f , D α F α β = J β e f f , G_(mu nu)+Lambdag_(mu nu)=8pi GT_(mu nu)^(eff),qquadD^( alpha)F_(alpha beta)=J_( beta)^(eff),\boxed{ G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G\,T^{\rm eff}_{\mu\nu},\qquad D^\alpha F_{\alpha\beta}=J^{\rm eff}_\beta, }Gμν+Λgμν=8πGTμνeff,DαFαβ=Jβeff,
with μ T μ ν e f f = 0 μ T μ ν e f f = 0 grad ^(mu)T_(mu nu)^(eff)=0\nabla^\mu T^{\rm eff}_{\mu\nu}=0μTμνeff=0 and D μ J μ e f f = 0 D μ J μ e f f = 0 D^( mu)J_( mu)^(eff)=0D^\mu J^{\rm eff}_\mu=0DμJμeff=0.
Proposition 5.4.3 (constants as multipliers = Γ-calibration).
Under Slater and strict convexity of F 0 F 0 F_(0)\mathcal F_0F0 along the cone-class, KKT multipliers ( G 1 , Λ , g Y M 2 ) ( G 1 , Λ , g Y M 2 ) (G^(-1),Lambda,g_(YM)^(-2))(G^{-1},\Lambda,g_{\rm YM}^{-2})(G1,Λ,gYM2) are unique. Calibrating F ε F ε F_(epsi)\mathcal F_\varepsilonFε on (i) Minkowski and (ii) constant-curvature backgrounds fixes the same constants; the Γ-calibrated constants coincide with the KKT duals.

5.5 — Fast-sector stationarity and microcausal hygiene

We price the derivation (the actual throughput) and solve the first-order condition on the unitary-orbit tangent using the Moore–Penrose pseudoinverse of the orbit Laplacian.

5.5.1 Unitary-orbit control with derivation-quadratic (Heisenberg law)

Work on a finite block Λ Λ Lambda\LambdaΛ with normalized Hilbert–Schmidt (HS) inner product X , Y H S := Tr Λ ( X Y ) / d Λ X , Y H S := Tr Λ ( X Y ) / d Λ (:X,Y:)_(HS):=Tr_(Lambda)(X^(†)Y)//d_( Lambda)\langle X,Y\rangle_{\rm HS}:=\mathrm{Tr}_\Lambda(X^\dagger Y)/d_\LambdaX,YHS:=TrΛ(XY)/dΛ. Let A t = U t A 0 U t A t = U t A 0 U t A_(t)=U_(t)^(†)A_(0)U_(t)A_t=U_t^\dagger A_0U_tAt=UtA0Ut, U ˙ t = i H t U t U ˙ t = i H t U t U^(˙)_(t)=-(i)/(ℏ)H_(t)U_(t)\dot U_t=-\tfrac{i}{\hbar}H_tU_tU˙t=iHtUt, so
A ˙ t = i [ H t , A t ] . A ˙ t = i [ H t , A t ] . A^(˙)_(t)=(i)/(ℏ)[H_(t),A_(t)].\boxed{\ \dot A_t=\tfrac{i}{\hbar}[H_t,A_t]\ }. A˙t=i[Ht,At] .
Let P ( A ) P ( A ) P(A)\mathsf P(A)P(A) be the orbit-restricted predictive score; G ( A ) := grad H S P ( A ) G ( A ) := grad H S P ( A ) G(A):=grad_(HS)P(A)G(A):=\operatorname{grad}_{\rm HS}\mathsf P(A)G(A):=gradHSP(A).
Key pairing (explicit HS calculation). Work on a finite block Λ Λ Lambda\LambdaΛ with normalized Hilbert–Schmidt inner product X , Y H S := Tr ( X Y ) / d Λ X , Y H S := Tr ( X Y ) / d Λ (:X,Y:)_(HS):=Tr(X^(†)Y)//d_( Lambda)\langle X,Y\rangle_{\rm HS}:=\mathrm{Tr}(X^\dagger Y)/d_\LambdaX,YHS:=Tr(XY)/dΛ. For Hermitian A , H A , H A,HA,HA,H,
[ A , X ] , Y H S = 1 d Λ Tr ( ( A X X A ) Y ) = 1 d Λ Tr ( X A Y A X Y ) = 1 d Λ Tr ( X A Y X Y A ) ( cyclicity ) = X , [ A , Y ] H S . [ A , X ] , Y H S = 1 d Λ Tr ( ( A X X A ) Y ) = 1 d Λ Tr ( X A Y A X Y ) = 1 d Λ Tr ( X A Y X Y A ) ( cyclicity ) = X , [ A , Y ] H S . {:[(:[A","X]","Y:)_(HS)=(1)/(d_( Lambda))Tr((AX-XA)^(†)Y)],[=(1)/(d_( Lambda))Tr(X^(†)AY-AX^(†)Y)],[=(1)/(d_( Lambda))Tr(X^(†)AY-X^(†)YA)quad("cyclicity")],[=(:X","[A","Y]:)_(HS).]:}\begin{aligned} \langle [A,X],Y\rangle_{\rm HS} &= \frac{1}{d_\Lambda}\mathrm{Tr}\big((AX-XA)^\dagger Y\big) \\ &= \frac{1}{d_\Lambda}\mathrm{Tr}\big(X^\dagger A Y - A X^\dagger Y\big) \\ &= \frac{1}{d_\Lambda}\mathrm{Tr}\big(X^\dagger A Y - X^\dagger Y A\big) \quad(\text{cyclicity})\\ &= \langle X,[A,Y]\rangle_{\rm HS}. \end{aligned}[A,X],YHS=1dΛTr((AXXA)Y)=1dΛTr(XAYAXY)=1dΛTr(XAYXYA)(cyclicity)=X,[A,Y]HS.
Thus the commutator superoperator ad A : X [ A , X ] ad A : X [ A , X ] ad_(A):X|->[A,X]\mathrm{ad}_A:X\mapsto[A,X]adA:X[A,X] is HS-self-adjoint on Hermitian A A AAA. Along a unitary orbit A t = U t A 0 U t A t = U t A 0 U t A_(t)=U_(t)^(†)A_(0)U_(t)A_t=U_t^\dagger A_0 U_tAt=UtA0Ut with U ˙ t = i H t U t U ˙ t = i H t U t U^(˙)_(t)=-(i)/(ℏ)H_(t)U_(t)\dot U_t=-\tfrac{i}{\hbar}H_tU_tU˙t=iHtUt,
A ˙ t = i [ H t , A t ] . A ˙ t = i [ H t , A t ] . A^(˙)_(t)=(i)/(ℏ)[H_(t),A_(t)].\dot A_t=\tfrac{i}{\hbar}[H_t,A_t].A˙t=i[Ht,At].
Let P ( A ) P ( A ) P(A)\mathsf P(A)P(A) be the orbit-restricted predictive score and G ( A ) := grad H S P ( A ) G ( A ) := grad H S P ( A ) G(A):=grad_(HS)P(A)G(A):=\operatorname{grad}_{\rm HS}\mathsf P(A)G(A):=gradHSP(A). For any Hermitian control H H HHH,
d d t | t = 0 P ( A t ) = G ( A ) , i [ H , A ] H S = i [ G ( A ) , A ] , H H S . d d t | t = 0 P ( A t ) = G ( A ) , i [ H , A ] H S = i [ G ( A ) , A ] , H H S . (d)/(dt)|_(t=0)P(A_(t))=(:G(A),(i)/(ℏ)[H,A]:)_(HS)=(:-(i)/(ℏ)[G(A),A],H:)_(HS).\frac{d}{dt}\Big|_{t=0}\mathsf P(A_t)=\Big\langle G(A),\tfrac{i}{\hbar}[H,A]\Big\rangle_{\rm HS} =\Big\langle -\tfrac{i}{\hbar}[G(A),A],\,H\Big\rangle_{\rm HS}.ddt|t=0P(At)=G(A),i[H,A]HS=i[G(A),A],HHS.
Hence the steepest-ascent Hamiltonian (Riesz/KKT on the unitary manifold) is
H ( A ) i [ G ( A ) , A ] , H ( A ) i [ G ( A ) , A ] , H^(**)(A)prop-(i)/(ℏ)[G(A),A],\boxed{\ H^*(A)\ \propto\ -\tfrac{i}{\hbar}[G(A),A]\ }, H(A)  i[G(A),A] ,
and the corresponding generator is A ˙ = i [ H ( A ) , A ] A ˙ = i [ H ( A ) , A ] A^(˙)=(i)/(ℏ)[H^(**)(A),A]\dot A=\tfrac{i}{\hbar}[H^*(A),A]A˙=i[H(A),A]. Local Lipschitz and blockwise bounds propagate to the quasi-local algebra via the uniform HS-block controls of Ch. 3.

5.5.2 Leakage and GKSL

Allow Lindblad controls { L j } { L j } {L_(j)}\{L_j\}{Lj} with leakage price λ l e a k j W 1 / 2 L j H S 2 λ l e a k j W 1 / 2 L j H S 2 lambda_(leak)sum _(j)||W^(1//2)L_(j)||_(HS)^(2)\lambda_{\rm leak}\sum_j\|W^{1/2}L_j\|_{\rm HS}^2λleakjW1/2LjHS2 (App. A). The convex optimization over CP-tangent directions yields an optimal GKSL generator with leakage-penalized pointer alignment as in Ch. 3. Hypotheses U give well-posedness and Lieb–Robinson-type bounds on the cone-class.

5.5.3 Microcausal hygiene

The poke cone, orbit locality, and LR-type bounds ensure commutator growth remains inside the cone; the principal-symbol lemma (§4.1) forbids order flips without paying the W 1 W 1 W_(1)W_1W1 gap. All variational steps above remain legal within the cone-preserving class.

5.6 — Summary (coupled laws)

At joint KKT stationarity of J J J\mathcal JJ on the cone-preserving class:
  • Slow (Einstein–Yang–Mills).
    G μ ν + Λ g μ ν = 8 π G T μ ν e f f , D α F α β = J β e f f , G μ ν + Λ g μ ν = 8 π G T μ ν e f f , D α F α β = J β e f f , G_(mu nu)+Lambdag_(mu nu)=8pi GT_(mu nu)^(eff),qquadD^( alpha)F_(alpha beta)=J_( beta)^(eff),G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G\,T^{\rm eff}_{\mu\nu},\qquad D^\alpha F_{\alpha\beta}=J^{\rm eff}_\beta,Gμν+Λgμν=8πGTμνeff,DαFαβ=Jβeff,
    with T e f f , J e f f T e f f , J e f f T^(eff),J^(eff)T^{\rm eff},J^{\rm eff}Teff,Jeff well-posed as Clarke subgradients (conserved for every selection).
    Under Assumption L (unique minimizer or Clarke-regular active set with constant directional derivatives), they upgrade to single linear distributions with distributional (Riesz-on-bounded-domains) representation, T μ ν e f f = T ν μ e f f T μ ν e f f = T ν μ e f f T_(mu nu)^(eff)=T_(nu mu)^(eff)T^{\rm eff}_{\mu\nu}=T^{\rm eff}_{\nu\mu}Tμνeff=Tνμeff, μ T μ ν e f f = 0 μ T μ ν e f f = 0 grad ^(mu)T_(mu nu)^(eff)=0\nabla^\mu T^{\rm eff}_{\mu\nu}=0μTμνeff=0, D μ J μ e f f = 0 D μ J μ e f f = 0 D^( mu)J_( mu)^(eff)=0D^\mu J^{\rm eff}_\mu=0DμJμeff=0.
  • Fast (unitary/GKSL).
    A ˙ = i [ H ( A ) , A ] , H ( A ) = A A + ( i [ A , G ( A ) ] ) , A A = ad A ad A = ad A 2 , = λ t h 1 . A ˙ = i [ H ( A ) , A ] , H ( A ) = A A + ( i [ A , G ( A ) ] ) , A A = ad A ad A = ad A 2 , = λ t h 1 . A^(˙)=(i)/(ℏ)[H^(**)(A),A],quadH^(**)(A)=A_(A)^(+)(-i[A,G(A)]),quadA_(A)=ad_(A)^(**)ad_(A)=ad_(A)^(2),quadℏ=lambda_(th)^(-1).\dot A=\tfrac{i}{\hbar}[H^*(A),A],\quad H^*(A)=\mathcal A_A^{+}\!\big(-i[A,G(A)]\big),\quad \mathcal A_A=\mathrm{ad}_A^{\,*}\mathrm{ad}_A=\mathrm{ad}_A^2,\quad \hbar=\lambda_{\rm th}^{-1}.A˙=i[H(A),A],H(A)=AA+(i[A,G(A)]),AA=adAadA=adA2,=λth1.
    With leakage, the optimal generator is GKSL with leakage-penalized pointer alignment; well-posed with LR-type bounds.
  • Constants. , G , Λ , g Y M , G , Λ , g Y M ℏ,G,Lambda,g_(YM)\hbar,\,G,\,\Lambda,\,g_{\rm YM},G,Λ,gYM are the unique KKT multipliers and match the Γ-calibrated constants (Prop. 5.4.3).
All steps are now rigorous and consistent with the coherence-budget framework: directional envelope calculus with direction-selecting minimizers; Clarke-sound sources and conservation (upgraded to tensors when linearity holds); first-variation convergence for the slow Γ-limit; and a derivation-quadratic control law on the unitary orbit (with pseudoinverse and explicit normalization) producing the Heisenberg/GKSL fast dynamics.

Chapter 6 — Gauge & Matter: Budget–Symmetry Selection; Hypercharge Uniqueness

Scope. With budgets and cone hygiene in place, we summarize the selection for the gauge scaffold and record the hypercharge‑uniqueness result under the standard binders (single Higgs doublet; anomaly cancellation; minimal chiral set). Proofs and linear‑system details mirror Appendix C.

6.1 Budget–symmetry selection (sketch)

Ad‑invariant quadratic forms on a compact simple Lie algebra are multiples of the Killing form; additivity across factors gives i C 2 ( G i ) i C 2 ( G i ) sum _(i)C_(2)(G_(i))\sum_i C_2(G_i)iC2(Gi). The unique Ad‑invariant count surcharge is i dim G i i dim G i sum _(i)dim G_(i)\sum_i\dim G_iidimGi. These, with Γ‑locality and mediator locality, generate the mediator part of the complexity budget B c x B c x B_(cx)B_{\rm cx}Bcx. Minimal coupling follows from the Γ‑limit of the connection scaffold.

6.2 Hypercharge uniqueness (binder set: Yukawa + anomalies + minimal set)

Assumptions for hypercharge (conditional uniqueness).
Uniqueness here is conditional on the binder set: (B_Yuk) renormalizable Yukawas, (B_{\rm anom}) anomaly cancellation, and (B_{\rm min}) the minimal chiral content with a single Higgs doublet. Relaxing any binder re-opens branches; see Appendix C for alternatives and their costs.
Binders. (i) B_Yuk: only renormalizable Yukawas Q H d c , Q H ~ u c , L H e c Q H d c , Q H ~ u c , L H e c QHd^(c),Q tilde(H)u^(c),LHe^(c)QH d^c,\ Q\tilde H u^c,\ L H e^cQHdc, QH~uc, LHec. (ii) B_anom: cancel [ S U ( 3 ) ] 2 U ( 1 ) Y , [ S U ( 2 ) ] 2 U ( 1 ) Y , U ( 1 ) Y 3 , grav 2 U ( 1 ) Y [ S U ( 3 ) ] 2 U ( 1 ) Y , [ S U ( 2 ) ] 2 U ( 1 ) Y , U ( 1 ) Y 3 , grav 2 U ( 1 ) Y [SU(3)]^(2)U(1)_(Y),[SU(2)]^(2)U(1)_(Y),U(1)_(Y)^(3),"grav"^(2)-U(1)_(Y)[SU(3)]^2U(1)_Y,\ [SU(2)]^2U(1)_Y,\ U(1)_Y^3,\ \text{grav}^2\!\!\!-\!U(1)_Y[SU(3)]2U(1)Y, [SU(2)]2U(1)Y, U(1)Y3, grav2U(1)Y anomalies. (iii) B_min: one‑generation minimal chiral set { Q , u c , d c , L , e c } { Q , u c , d c , L , e c } {Q,u^(c),d^(c),L,e^(c)}\{Q,u^c,d^c,L,e^c\}{Q,uc,dc,L,ec}; one Higgs doublet.
Statement. Let ( Y Q , Y u , Y d , Y L , Y e , Y H ) ( Y Q , Y u , Y d , Y L , Y e , Y H ) (Y_(Q),Y_(u),Y_(d),Y_(L),Y_(e),Y_(H))(Y_Q,Y_u,Y_d,Y_L,Y_e,Y_H)(YQ,Yu,Yd,YL,Ye,YH) be unknown hypercharges. Under B_Yuk + B_anom + B_min, the solution set is a one‑parameter line (overall normalization/orientation). Fixing the unit with the minimal‑charge binder ( Q = T 3 + Y Q = T 3 + Y Q=T_(3)+YQ=T_3+YQ=T3+Y and color‑singlet integrality) yields the Standard Model values
( Y Q , Y u , Y d , Y L , Y e , Y H ) = ( 1 6 , 2 3 , 1 3 , 1 2 , 1 , 1 2 ) . ( Y Q , Y u , Y d , Y L , Y e , Y H ) = ( 1 6 , 2 3 , 1 3 , 1 2 , 1 , 1 2 ) . (Y_(Q),Y_(u),Y_(d),Y_(L),Y_(e),Y_(H))=((1)/(6),-(2)/(3),(1)/(3),-(1)/(2),1,-(1)/(2)).\boxed{(Y_Q,Y_u,Y_d,Y_L,Y_e,Y_H)=\big(\tfrac16,-\tfrac23,\tfrac13,-\tfrac12,1,-\tfrac12\big)}.(YQ,Yu,Yd,YL,Ye,YH)=(16,23,13,12,1,12).
Proof (linear system & rank). Yukawa invariance gives Y Q + Y H + Y d = 0 Y Q + Y H + Y d = 0 Y_(Q)+Y_(H)+Y_(d)=0Y_Q+Y_H+Y_d=0YQ+YH+Yd=0, Y Q Y H + Y u = 0 Y Q Y H + Y u = 0 Y_(Q)-Y_(H)+Y_(u)=0Y_Q-Y_H+Y_u=0YQYH+Yu=0, Y L + Y H + Y e = 0 Y L + Y H + Y e = 0 Y_(L)+Y_(H)+Y_(e)=0Y_L+Y_H+Y_e=0YL+YH+Ye=0. Anomalies add 3 Y Q + Y L = 0 3 Y Q + Y L = 0 3Y_(Q)+Y_(L)=03Y_Q+Y_L=03YQ+YL=0 and 6 Y Q + 3 Y u + 3 Y d + 2 Y L + Y e = 0 6 Y Q + 3 Y u + 3 Y d + 2 Y L + Y e = 0 6Y_(Q)+3Y_(u)+3Y_(d)+2Y_(L)+Y_(e)=06Y_Q+3Y_u+3Y_d+2Y_L+Y_e=06YQ+3Yu+3Yd+2YL+Ye=0. Solve: Y H = 3 Y Q Y H = 3 Y Q Y_(H)=-3Y_(Q)Y_H=-3Y_QYH=3YQ, Y L = 3 Y Q Y L = 3 Y Q Y_(L)=-3Y_(Q)Y_L=-3Y_QYL=3YQ, Y e = + 6 Y Q Y e = + 6 Y Q Y_(e)=+6Y_(Q)Y_e=+6Y_QYe=+6YQ, Y u = 4 Y Q Y u = 4 Y Q Y_(u)=-4Y_(Q)Y_u=-4Y_QYu=4YQ, Y d = + 2 Y Q Y d = + 2 Y Q Y_(d)=+2Y_(Q)Y_d=+2Y_QYd=+2YQ with Y Q Y Q Y_(Q)Y_QYQ free (rank 5/6). Minimal‑charge normalization via Q = T 3 + Y Q = T 3 + Y Q=T_(3)+YQ=T_3+YQ=T3+Y and Q ( ν ) = 0 Q ( ν ) = 0 Q(nu)=0Q(\nu)=0Q(ν)=0 fixes Y Q = 1 / 6 Y Q = 1 / 6 Y_(Q)=1//6Y_Q=1/6YQ=1/6. After normalization, strict convexity/tie‑breakers on B c x B c x B_(cx)B_{\rm cx}Bcx exclude co‑minima; the SM assignment is unique with a positive gap.

Chapter 7 — Horizons: Pointer Mechanics ⇒ Area Law; Hawking Flux Suppression (Repaired & Airtight)

Scope. We work in a local Killing-horizon patch with surface gravity $\kappa>0$, the single-cone class and the C*-compatible budgets of Ch. 3, Γ-compact slow sector of Ch. 4, and the coupled-laws framework of Ch. 5. We:
(i) specify a near-horizon Unruh-diagonal GKSL model fixed by pointer alignment;
(ii) take mutual information $I(A:\bar A)$ as the primary entropy functional (with $E_R\le I$), prove quasi-factorization with explicit LR-dependent constants;
(iii) give a per-tile information–budget bound controlled by the leakage budget; define a pointer cutoff $\ell_*$ by KKT thresholding; and prove an area-law upper bound with a calibrated constant;
(iv) prove a sharp flux-suppression inequality as a linear program in the rates, in full generality and with an exact Hawking-weight corollary;
(v) record microcausality and an identifiability lemma to determine Hawking rates from pointer-basis two-point data.
Proof details and constant tracking are in Appendix G (area law) and Appendix H (flux suppression).
KPI & refuter. The near-horizon prediction is a universal amplitude suppression of the outgoing flux by a budget-controlled factor at fixed Hawking temperature. Any observed temperature shift at leading order would falsify the budget-consistent GKSL picture; amplitude-only suppression with the predicted frequency shaping supports it.

7.1 Near-horizon setup, pointer alignment, and quasi-factorization

7.1.1 Geometry and Unruh modes

Fix a stationary Killing horizon with surface gravity $\kappa>0$. On a bounded chart of the horizon neighborhood we use Rindler coordinates to the accuracy guaranteed by Ch. 4 bounded-geometry hypotheses. Let
{ b k o u t , b k i n } k K { b k o u t , b k i n } k K {b_(k)^(out),b_(k)^(in)}_(k inK)\{b_k^{\rm out},\,b_k^{\rm in}\}_{k\in\mathcal K}{bkout,bkin}kK
denote Unruh mode operators localized to the patch (frequency $\omega_k>0$ and tangential quantum numbers). The Unruh temperature is
T U = κ 2 π , β U := T U 1 . T U = κ 2 π , β U := T U 1 . T_(U)=(kappa)/(2pi),qquadbeta _(U):=T_(U)^(-1).T_U=\frac{\kappa}{2\pi},\qquad \beta_U:=T_U^{-1}.TU=κ2π,βU:=TU1.

7.1.2 Fast-sector GKSL and pointer alignment

On a horizon tile $T$ (area $|T|$), the fast sector evolves by a GKSL generator that is diagonal in the Unruh (pointer) basis:
L T ( ρ ) = k K T [ γ k D [ b k o u t ] ( ρ ) + γ ~ k D [ ( b k o u t ) ] ( ρ ) ] , D [ C ] ( ρ ) := C ρ C 1 2 { C C , ρ } . L T ( ρ ) = k K T [ γ k D [ b k o u t ] ( ρ ) + γ ~ k D [ ( b k o u t ) ] ( ρ ) ] , D [ C ] ( ρ ) := C ρ C 1 2 { C C , ρ } . L_(T)(rho)=sum_(k inK_(T))[gamma _(k)D[b_(k)^(out)](rho)+ tilde(gamma)_(k)D[(b_(k)^(out))^(†)](rho)],quadD[C](rho):=C rhoC^(†)-(1)/(2){C^(†)C,rho}.\mathcal L_T(\rho)=\sum_{k\in\mathcal K_T}\!\big[\gamma_k\,\mathcal D[b_k^{\rm out}](\rho)+\tilde\gamma_k\,\mathcal D[(b_k^{\rm out})^\dagger](\rho)\big],\quad \mathcal D[C](\rho):=C\rho C^\dagger-\tfrac12\{C^\dagger C,\rho\}.LT(ρ)=kKT[γkD[bkout](ρ)+γ~kD[(bkout)](ρ)],D[C](ρ):=CρC12{CC,ρ}.
The leakage weight $W\succ0$ (Appendix A) co-diagonalizes with the Unruh basis; write $w_k:=\langle k|W|k\rangle>0$. The per-tile leakage budget is
B l e a k ( T ) = k K T w k ( γ k + γ ~ k ) . B l e a k ( T ) = k K T w k ( γ k + γ ~ k ) . B_(leak)(T)=sum_(k inK_(T))w_(k)(gamma _(k)+ tilde(gamma)_(k)).\mathcal B_{\rm leak}(T)=\sum_{k\in\mathcal K_T} w_k\,(\gamma_k+\tilde\gamma_k).Bleak(T)=kKTwk(γk+γ~k).
Hawking detailed balance has
γ ~ k H = γ k H e β U ω k , n ¯ k = ( e β U ω k 1 ) 1 . γ ~ k H = γ k H e β U ω k , n ¯ k = ( e β U ω k 1 ) 1 . tilde(gamma)_(k)^(H)=gamma_(k)^(H)e^(-beta _(U)omega _(k)),qquad bar(n)_(k)=(e^(beta _(U)omega _(k))-1)^(-1).\tilde\gamma_k^{\rm H}=\gamma_k^{\rm H}\,e^{-\beta_U\omega_k},\qquad \bar n_k=(e^{\beta_U\omega_k}-1)^{-1}.γ~kH=γkHeβUωk,n¯k=(eβUωk1)1.
Lemma 7.1 (Budget-monotonic pointer alignment).
Let L T L T L_(T)\mathcal{L}_TLT be any GKSL on T T TTT with the same singular values of the noise block as above. Let Δ Δ Delta\DeltaΔ be the pinching (full dephasing) in the Unruh basis. Then:
  1. (Budget) j ω ( L j W L j ) j ω ( ( Δ L j ) W ( Δ L j ) ) j ω ( L j W L j ) j ω ( ( Δ L j ) W ( Δ L j ) ) sum _(j)omega(L_(j)^(†)WL_(j)) >= sum _(j)omega((DeltaL_(j))^(†)W(DeltaL_(j)))\sum_j \omega(L_j^\dagger W L_j) \ge \sum_j \omega \big( (\Delta L_j)^\dagger W (\Delta L_j) \big)jω(LjWLj)jω((ΔLj)W(ΔLj)) (Hilbert-Schmidt pinching contraction).
  2. (Information) For any state ρ ρ rho\rhoρ, I ( ( id Λ t ) ( ρ ) : T ¯ ) I ( ( id Λ t ) ( ρ ) : T ¯ ) I((idoxLambda _(t))(rho): bar(T))I\big( (\mathrm{id} \otimes \Lambda_t)(\rho) : \bar{T} \big)I((idΛt)(ρ):T¯) does not increase if one replaces Λ t := e t L T Λ t := e t L T Lambda _(t):=e^(tL_(T))\Lambda_t := e^{t \mathcal{L}_T}Λt:=etLT by Δ Λ t Δ Δ Λ t Δ Delta@Lambda _(t)@Delta\Delta \circ \Lambda_t \circ \DeltaΔΛtΔ (monotonicity of I I III under local CPTP maps and commutation of Δ Δ Delta\DeltaΔ with the Unruh-diagonal semigroup).
Hence, among channels with fixed spectral data, Unruh-diagonal noise minimizes leakage cost and maximizes I ( T : T ) I ( T : T ¯ ) I(T: bar(T))I(T : \overline{T})I(T:T) only within that spectral class. We therefore restrict to the diagonal form without loss for upper bounds.
Proof. (1) The map X W 1 / 2 X X W 1 / 2 X X|->W^(1//2)XX \mapsto W^{1/2} XXW1/2X followed by conditional expectation Δ Δ Delta\DeltaΔ is a contraction in the HS norm; sum over j j jjj. (2) Mutual information is monotone under local CPTP maps; Δ Δ Delta\DeltaΔ is local on T T TTT and leaves the Unruh-diagonal dynamics invariant; compose. \square

7.1.3 Entropy functional and the $E_R$ relation

We take mutual information as primary:
I ( A : A ¯ ) = D ( ρ A A ¯ ρ A ρ A ¯ ) . I ( A : A ¯ ) = D ( ρ A A ¯ ρ A ρ A ¯ ) . I(A: bar(A))=D(rho_(A bar(A))||rho _(A)oxrho_( bar(A))).I(A:\bar A)=D\big(\rho_{A\bar A}\,\big\|\,\rho_A\otimes\rho_{\bar A}\big).I(A:A¯)=D(ρAA¯ρAρA¯).
We use the general inequality
E R ( ρ A A ¯ ) I ( A : A ¯ ) E R ( ρ A A ¯ ) I ( A : A ¯ ) E_(R)(rho_(A bar(A))) <= I(A: bar(A))\boxed{\,E_R(\rho_{A\bar A})\ \le\ I(A:\bar A)\,}ER(ρAA¯)  I(A:A¯)
(no $\tfrac12$ factor in general). All subsequent bounds are proved for $I$ and then immediately transfer to $E_R$.

7.1.4 Quasi-factorization with LR constants

Tile a region A A AAA by disjoint squares { T j } j J ( A ) { T j } j J ( A ) {T_(j)}_(j in J(A))\{T_j\}_{j\in J(A)}{Tj}jJ(A) of side \ell, and include a boundary collar of width O ( ) O ( ) O(ℓ)O(\ell)O(), yielding O ( | A | ) O ( | A | ) O(|del A|)O(|\partial A|)O(|A|) collar tiles. The GKSL semigroup obeys cone-limited LR bounds (Ch. 3 U4; Appendix F.2) and admits an Unruh thermal log-Sobolev (or spectral-gap) constant on each tile (Appendix G.1).
Theorem 7.0 (Quasi-factorization of mutual information).
There exist C L R , ξ , v L R < C L R , ξ , v L R < C_(LR),xi,v_(LR) < ooC_{\rm LR}, \xi, v_{\rm LR} < \inftyCLR,ξ,vLR<, depending only on the LR/mixing data and the local mode density, such that for all 0 0 ℓ >= ℓ_(0)\ell \ge \ell_00 and all horizon-patch regions A A AAA,
I ( A : A ¯ ) j J ( A ) I ( T j : T j ) + C L R | A | + C L R e / ξ . I ( A : A ¯ ) j J ( A ) I ( T j : T j ¯ ) + C L R | A | + C L R e / ξ . I(A: bar(A)) <= sum_(j in J(A))I(T_(j): bar(T_(j)))+C_(LR)|del A|+C_(LR)e^(-ℓ//xi).\boxed{ I(A : \bar{A}) \le \sum_{j \in J(A)} I(T_j : \overline{T_j}) + C_{\rm LR} \, |\partial A| + C_{\rm LR} \, e^{-\ell / \xi}. }I(A:A¯)jJ(A)I(Tj:Tj)+CLR|A|+CLRe/ξ.
In particular, choosing c , ξ log ( 1 + | A | ) c , ξ log ( 1 + | A | ) ℓ >= c,xi log(1+|del A|)\ell \ge c , \xi \log(1 + |\partial A|)c,ξlog(1+|A|) absorbs the remainder into the boundary term: I ( A : A ¯ ) j I ( T j : T j ) + O ( | A | ) I ( A : A ¯ ) j I ( T j : T j ¯ ) + O ( | A | ) I(A: bar(A)) <= sum _(j)I(T_(j): bar(T_(j)))+O(|del A|)I(A : \bar{A}) \le \sum_j I(T_j : \overline{T_j}) + O(|\partial A|)I(A:A¯)jI(Tj:Tj)+O(|A|).
Proof (outline; Appendix G.2). Chain the mutual information by SSA:
I ( A : A ¯ ) = j I ( T j : A ¯ , | T < j ) I ( A : A ¯ ) = j I ( T j : A ¯ , | T < j ) I(A: bar(A))=sum _(j)I(T_(j): bar(A),|T_( < j))I(A : \bar{A}) = \sum_j I \big( T_j : \bar{A} , \big| T_{<j} \big)I(A:A¯)=jI(Tj:A¯,|T<j) and bound each conditional term by an LR-decaying influence from T j T j ¯ bar(T_(j))\overline{T_j}Tj outside a collar of width ∼ℓ\sim \ell. Mix to Unruh steady on the collar using the tile log-Sobolev gap; constants track to C L R , ξ C L R , ξ C_(LR),xiC_{\rm LR}, \xiCLR,ξ. \square

7.2 Area-law upper bound with calibrated constant

We now reduce the area bound to a per-tile estimate and calibrate the constant.

7.2.1 Per-tile information vs. leakage budget

For a tile T T TTT, write Γ k := γ k + γ ~ k Γ k := γ k + γ ~ k Gamma _(k):=gamma _(k)+ tilde(gamma)_(k)\Gamma_k := \gamma_k + \tilde{\gamma}_kΓk:=γk+γ~k and define
τ l e a k ( T ) := k K T w k Γ k . τ l e a k ( T ) := k K T w k Γ k . tau_(leak)(T):=sum_(k inK_(T))w_(k)Gamma _(k).\tau_{\rm leak}(T) := \sum_{k \in \mathcal{K}_T} w_k \, \Gamma_k.τleak(T):=kKTwkΓk.
Proposition 7.1 (Linear upper bound, sharp coefficient).
There exists a finite constant
χ T := sup k K T + I k Γ k 1 w k , χ T := sup k K T + I k Γ k 1 w k , chi _(T):=s u p_(k inK_(T))(del^(+)I_(k))/(delGamma _(k))(1)/(w_(k)),\chi_T := \sup_{k \in \mathcal{K}_T} \ \frac{\partial^+ I_k}{\partial \Gamma_k} \, \frac{1}{w_k} \, ,χT:=supkKT +IkΓk1wk,
(where I k I k I_(k)I_kIk is the single-mode contribution under Unruh-diagonal dynamics and + + del^(+)\partial^++ is the right derivative at the realized Γ k Γ k Gamma _(k)\Gamma_kΓk) such that
I ( T : T ) χ T τ l e a k ( T ) . I ( T : T ¯ ) χ T τ l e a k ( T ) . I(T: bar(T)) <= chi _(T)tau_(leak)(T).\boxed{ I(T : \overline{T}) \le \chi_T \, \tau_{\rm leak}(T) \ . }I(T:T)χTτleak(T) .
Moreover χ T χ T chi _(T)\chi_TχT depends only on the Unruh temperature, the LR/mixing constants, and the local mode density; it is uniform across tiles of the same side \ell.
Proof (Appendix G.3). For Unruh-diagonal GKSL, I ( T : T ) = k I k ( Γ k ) I ( T : T ¯ ) = k I k ( Γ k ) I(T: bar(T))=sum _(k)I_(k)(Gamma _(k))I(T : \overline{T}) = \sum_k I_k(\Gamma_k)I(T:T)=kIk(Γk). Each I k I k I_(k)I_kIk is concave, increasing in Γ k Γ k Gamma _(k)\Gamma_kΓk and differentiable a.e. (data processing + semigroup contractivity in the normalized-HS metric used for budgets, Appendix E.1). Lagrange’s inequality then gives
I ( T : T ) k ( + I k / Γ k ) Γ k χ T k w k Γ k . I ( T : T ¯ ) k ( + I k / Γ k ) Γ k χ T k w k Γ k . I(T: bar(T)) <= sum _(k)(del^(+)I_(k)//delGamma _(k))Gamma _(k) <= chi _(T)sum _(k)w_(k)Gamma _(k).I(T : \overline{T}) \le \sum_k (\partial^+ I_k / \partial \Gamma_k) \Gamma_k \le \chi_T \sum_k w_k \Gamma_k.I(T:T)k(+Ik/Γk)ΓkχTkwkΓk.
Uniformity follows from bounded geometry and the common Unruh temperature. \square
Remark. We carry (7.2.1) as an upper bound. Under the Hawking-weight calibration of §7.3.2 the coefficient becomes tile-independent ( χ T χ χ T χ chi _(T)-=chi_(**)\chi_T \equiv \chi_*χTχ) and the bound is tight (equality along the Hawking ray).

7.2.2 KKT thresholding and the pointer cutoff $\ell_*$

We optimize per-tile I ( T : T ) I ( T : T ¯ ) I(T: bar(T))I(T : \overline{T})I(T:T) subject to the leakage allowance τ l e a k ( T ) τ l e a k ( T ) tau_(leak)(T)\tau_{\rm leak}(T)τleak(T). The KKT conditions with multiplier λ l e a k > 0 λ l e a k > 0 lambda_(leak) > 0\lambda_{\rm leak} > 0λleak>0 give the threshold rule
+ I k Γ k λ l e a k w k , Γ k > 0 I k Γ k = λ l e a k w k . + I k Γ k λ l e a k w k , Γ k > 0 I k Γ k = λ l e a k w k . (del^(+)I_(k))/(delGamma _(k)) <= lambda_(leak)w_(k),quadGamma _(k) > 0=>(delI_(k))/(delGamma _(k))=lambda_(leak)w_(k).\frac{\partial^+ I_k}{\partial \Gamma_k} \le \lambda_{\rm leak} \, w_k, \quad \Gamma_k > 0 \ \Rightarrow \ \frac{\partial I_k}{\partial \Gamma_k} = \lambda_{\rm leak} \, w_k.+IkΓkλleakwk,Γk>0  IkΓk=λleakwk.
Because w k w k w_(k)w_kwk grows monotonically with transverse momentum (pointer/energy scaling) and I k / Γ k I k / Γ k delI_(k)//delGamma _(k)\partial I_k / \partial \Gamma_kIk/Γk is bounded and decreases with | k | | k | |k||k||k| at fixed β U β U beta _(U)\beta_UβU, the optimizer activates modes only for | k | 1 | k | 1 |k|≲ℓ_(**)^(-1)|k| \lesssim \ell_*^{-1}|k|1 with a sharp cutoff at
= ( λ l e a k , W ) defined by I k Γ k | | k | = 1 = λ l e a k w k . = ( λ l e a k , W ) defined by I k Γ k | | k | = 1 = λ l e a k w k . ℓ_(**)=ℓ_(**)(lambda_(leak),W)"defined by"(delI_(k))/(delGamma_(k))|_(|k|=ℓ_(**)^(-1))=lambda_(leak)w_(k).\boxed{ \ell_* = \ell_*(\lambda_{\rm leak}, W) \ \ \text{defined by} \ \ \frac{\partial I_{k}}{\partial \Gamma_{k}} \Big|_{|k| = \ell_*^{-1}} = \lambda_{\rm leak} \, w_{k} \ . }=(λleak,W)  defined by  IkΓk||k|=1=λleakwk .
(Details in Appendix G.4.) For tiles with side = Θ ( ) = Θ ( ) ℓ=Theta(ℓ_(**))\ell = \Theta(\ell_*)=Θ(), the number of active transverse modes is | T | / d 2 | T | / d 2 ≃|T|//ℓ_(**)^(d-2)\simeq |T| / \ell_*^{d-2}|T|/d2 up to boundary/collar corrections.

7.2.3 Area-law theorem

Theorem 7.1 (Area-law upper bound with boundary term).
For tiles of side = Θ ( ) = Θ ( ) ℓ=Theta(ℓ_(**))\ell = \Theta(\ell_*)=Θ(),
I ( A : A ¯ ) χ ( β U , L R , W ) A r e a ( A ) d 2 + O ( | A | ) , E R ( A : A ¯ ) I ( A : A ¯ ) . I ( A : A ¯ ) χ ( β U , L R , W ) A r e a ( A ) d 2 + O ( | A | ) , E R ( A : A ¯ ) I ( A : A ¯ ) . I(A: bar(A)) <= chi_(**)(beta _(U),LR,W)(Area(A))/(ℓ_(**)^(d-2))+O(|del A|),qquadE_(R)(A: bar(A)) <= I(A: bar(A)).\boxed{ I(A : \bar{A}) \le \chi_*(\beta_U, {\rm LR}, W) \, \frac{{\rm Area}(A)}{\ell_*^{d-2}} + O(|\partial A|), \qquad E_R(A : \bar{A}) \le I(A : \bar{A}). }I(A:A¯)χ(βU,LR,W)Area(A)d2+O(|A|),ER(A:A¯)I(A:A¯).
Here χ := sup T χ T χ := sup T χ T chi_(**):=s u p _(T)chi _(T)\chi_* := \sup_T \chi_Tχ:=supTχT is finite and depends only on the Unruh temperature, LR/mixing constants and the local weight profile W W WWW (Appendix G.5). The O ( | A | ) O ( | A | ) O(|del A|)O(|\partial A|)O(|A|) constant depends only on LR/geometry data.
Proof. Combine Theorem 7.0 with Proposition 7.1 and the thresholding definition of ℓ_(**)\ell_*; count active modes per interior tile and absorb LR remainders into the collar. \square

7.2.4 Calibration to Einstein–Hilbert and the 1 / 4 1 / 4 1//41/41/4 coefficient

Proposition 7.2 (EH calibration =>\Rightarrow χ = 1 4 χ = 1 4 chi_(**)=(1)/(4)\chi_* = \tfrac{1}{4}χ=14 in Planck units).
Under the Γ-limit normalization of Appendix D (Ch. 4/D.5), the slow action equals Einstein–Hilbert and the Unruh temperature is fixed by κ κ kappa\kappaκ. Matching the tile-wise Clausius relation δ Q = T U δ S δ Q = T U δ S delta Q=T_(U)delta S\delta Q = T_U \delta SδQ=TUδS with the leakage work priced by W W WWW at the pointer cutoff yields
χ = 1 4 , I ( A : A ¯ ) A r e a ( A ) 4 P d 2 + O ( | A | ) . χ = 1 4 , I ( A : A ¯ ) A r e a ( A ) 4 P d 2 + O ( | A | ) . chi_(**)=(1)/(4),qquad I(A: bar(A)) <= (Area(A))/(4ℓ_(P)^(d-2))+O(|del A|).\boxed{ \chi_* = \frac{1}{4} , \qquad I(A : \bar{A}) \le \frac{{\rm Area}(A)}{4 \, \ell_P^{d-2}} + O(|\partial A|). }χ=14,I(A:A¯)Area(A)4Pd2+O(|A|).
Proof (Appendix G.6). The per-tile leakage expenditure that realizes the Unruh KMS structure at the cutoff equals the heat δ Q δ Q delta Q\delta QδQ through the stretched horizon; with T U T U T_(U)T_UTU fixed, the δ S δ S delta S\delta SδS density matches EH’s area-entropy density, fixing χ = 1 / 4 χ = 1 / 4 chi_(**)=1//4\chi_* = 1/4χ=1/4. \square

7.3 Hawking-flux suppression: sharp LP bound and calibrated equality

7.3.1 General linear-program bound (no profile assumptions)

Let c k c k c_(k)c_kck denote the outgoing flux weight per mode (number or energy),
c k := { v k n ¯ k (number flux) , ω k v k n ¯ k (energy flux) , 0 < v k v max . c k := v k n ¯ k      (number flux) , ω k v k n ¯ k      (energy flux) , 0 < v k v max . c_(k):={[v_(k) bar(n)_(k),(number flux)","],[ℏomega _(k)v_(k) bar(n)_(k),(energy flux)","]:}qquad0 < v_(k) <= v_(max).c_k := \begin{cases} v_k \, \bar{n}_k & \text{(number flux)}, \\[2pt] \hbar \omega_k \, v_k \, \bar{n}_k & \text{(energy flux)}, \end{cases} \qquad 0 < v_k \le v_{\max}.ck:={vkn¯k(number flux),ωkvkn¯k(energy flux),0<vkvmax.
For a tile T T TTT,
F o u t ( T ) = k K T c k Γ k , τ l e a k ( T ) = k K T w k Γ k . F o u t ( T ) = k K T c k Γ k , τ l e a k ( T ) = k K T w k Γ k . F_(out)(T)=sum_(k inK_(T))c_(k)Gamma _(k),qquadtau_(leak)(T)=sum_(k inK_(T))w_(k)Gamma _(k).\mathcal{F}_{\rm out}(T) = \sum_{k \in \mathcal{K}_T} c_k \, \Gamma_k, \qquad \tau_{\rm leak}(T) = \sum_{k \in \mathcal{K}_T} w_k \, \Gamma_k.Fout(T)=kKTckΓk,τleak(T)=kKTwkΓk.
Theorem 7.2 (General LP suppression).
For every tile T T TTT,
F o u t ( T ) ( max k K T c k w k ) τ l e a k ( T ) . F o u t ( T ) ( max k K T c k w k ) τ l e a k ( T ) . F_(out)(T) <= (max_(k inK_(T))(c_(k))/(w_(k)))tau_(leak)(T).\boxed{ \mathcal{F}_{\rm out}(T) \le \Big( \max_{k \in \mathcal{K}_T} \frac{c_k}{w_k} \Big) \tau_{\rm leak}(T) \ . }Fout(T)(maxkKTckwk)τleak(T) .
Equality holds by concentrating the budget on a mode with maximal ratio c k / w k c k / w k c_(k)//w_(k)c_k / w_kck/wk. Summing tiles gives
F o u t ( max k c k w k ) τ l e a k . F o u t ( max k c k w k ) τ l e a k . F_(out) <= (max_(k)(c_(k))/(w_(k)))tau_(leak).\boxed{ \mathcal{F}_{\rm out} \le \Big( \max_{k} \frac{c_k}{w_k} \Big) \tau_{\rm leak} \ . }Fout(maxkckwk)τleak .
Proof. Linear objective over a simplex: maximize c k Γ k c k Γ k sumc_(k)Gamma _(k)\sum c_k \Gamma_kckΓk s.t. w k Γ k τ w k Γ k τ sumw_(k)Gamma _(k) <= tau\sum w_k \Gamma_k \le \tauwkΓkτ and Γ k 0 Γ k 0 Gamma _(k) >= 0\Gamma_k \ge 0Γk0. KKT gives support on argmax of c k / w k c k / w k c_(k)//w_(k)c_k / w_kck/wk. \square
Microcausality. The LR bounds (Appendix F.2) imply
[ Φ t ( O X ) , O Y ] C e ( d ( X , Y ) v L R t ) / ξ [ Φ t ( O X ) , O Y ] C e ( d ( X , Y ) v L R t ) / ξ ||[Phi _(t)(O_(X)),O_(Y)]|| <= Ce^(-(d(X,Y)-v_(LR)t)//xi)\| [\Phi_t(O_X), O_Y] \| \le C e^{-(d(X,Y) - v_{\rm LR} t)/\xi}[Φt(OX),OY]Ce(d(X,Y)vLRt)/ξ; hence no superluminal contribution to F o u t F o u t F_(out)\mathcal{F}_{\rm out}Fout is admissible.

7.3.2 Hawking-weight calibration and exact multiplicative factor

Define the Hawking-weight W H W H W_(H)W_{\rm H}WH by
w k H c k , w k H c k , w_(k)^(H)propc_(k),\boxed{ w_k^{\rm H} \propto c_k \ , }wkHck ,
i.e. choose the leakage weight to price exactly the flux contribution (number or energy). This choice is canonical operationally (Appendix J.1: weight normalization by KPI).
Let Γ k H Γ k H Gamma_(k)^(H)\Gamma_k^{\rm H}ΓkH be the detailed-balance rates reproducing the semiclassical Hawking flux, and define the Hawking leakage cost
τ H ( T ) := k K T w k H Γ k H , F H a w k i n g ( T ) := k c k Γ k H . τ H ( T ) := k K T w k H Γ k H , F H a w k i n g ( T ) := k c k Γ k H . tau_(H)(T):=sum_(k inK_(T))w_(k)^(H)Gamma_(k)^(H),qquadF_(Hawking)(T):=sum _(k)c_(k)Gamma_(k)^(H).\tau_{\rm H}(T) := \sum_{k \in \mathcal{K}_T} w_k^{\rm H} \, \Gamma_k^{\rm H}, \qquad \mathcal{F}_{\rm Hawking}(T) := \sum_k c_k \, \Gamma_k^{\rm H}.τH(T):=kKTwkHΓkH,FHawking(T):=kckΓkH.
By construction,
F H a w k i n g ( T ) τ H ( T ) = k c k Γ k H k w k H Γ k H = k c k Γ k H k ( α c k ) Γ k H = α 1 , F H a w k i n g ( T ) τ H ( T ) = k c k Γ k H k w k H Γ k H = k c k Γ k H k ( α c k ) Γ k H = α 1 , (F_(Hawking)(T))/(tau_(H)(T))=(sum _(k)c_(k)Gamma_(k)^(H))/(sum _(k)w_(k)^(H)Gamma_(k)^(H))=(sum _(k)c_(k)Gamma_(k)^(H))/(sum _(k)(alphac_(k))Gamma_(k)^(H))=alpha^(-1),\frac{\mathcal{F}_{\rm Hawking}(T)}{\tau_{\rm H}(T)} = \frac{\sum_k c_k \Gamma_k^{\rm H}}{\sum_k w_k^{\rm H} \Gamma_k^{\rm H}} = \frac{\sum_k c_k \Gamma_k^{\rm H}}{\sum_k (\alpha c_k) \Gamma_k^{\rm H}} = \alpha^{-1},FHawking(T)τH(T)=kckΓkHkwkHΓkH=kckΓkHk(αck)ΓkH=α1,
a mode-independent constant (absorbed into the normalization of W H W H W_(H)W_{\rm H}WH).
Corollary 7.2′ (Budget-limited Hawking).
With W = W H W = W H W=W_(H)W = W_{\rm H}W=WH, the LP optimizer aligns with the Hawking ray and yields
F o u t ( T ) = min { 1 , τ l e a k ( T ) τ H ( T ) } F H a w k i n g ( T ) , F o u t ( T ) = min { 1 , τ l e a k ( T ) τ H ( T ) } F H a w k i n g ( T ) , F_(out)(T)=min{1,(tau_(leak)(T))/(tau_(H)(T))}F_(Hawking)(T),\boxed{ \mathcal{F}_{\rm out}(T) = \min \! \Big\{ 1, \ \frac{\tau_{\rm leak}(T)}{\tau_{\rm H}(T)} \Big\} \mathcal{F}_{\rm Hawking}(T), }Fout(T)=min{1, τleak(T)τH(T)}FHawking(T),
with equality cases: (i) τ l e a k ( T ) τ H ( T ) τ l e a k ( T ) τ H ( T ) tau_(leak)(T) >= tau_(H)(T)\tau_{\rm leak}(T) \ge \tau_{\rm H}(T)τleak(T)τH(T) and Γ k = Γ k H Γ k = Γ k H Gamma _(k)=Gamma_(k)^(H)\Gamma_k = \Gamma_k^{\rm H}Γk=ΓkH (full Hawking), (ii) τ l e a k ( T ) < τ H ( T ) τ l e a k ( T ) < τ H ( T ) tau_(leak)(T) < tau_(H)(T)\tau_{\rm leak}(T) < \tau_{\rm H}(T)τleak(T)<τH(T) and Γ k = α Γ k H Γ k = α Γ k H Gamma _(k)=alphaGamma_(k)^(H)\Gamma_k = \alpha \Gamma_k^{\rm H}Γk=αΓkH with α = τ l e a k ( T ) / τ H ( T ) α = τ l e a k ( T ) / τ H ( T ) alpha=tau_(leak)(T)//tau_(H)(T)\alpha = \tau_{\rm leak}(T) / \tau_{\rm H}(T)α=τleak(T)/τH(T) (budget-limited Hawking). Summing tiles gives
F o u t = min { 1 , τ l e a k τ H ( W H ) } F H a w k i n g . F o u t = min { 1 , τ l e a k τ H ( W H ) } F H a w k i n g . F_(out)=min{1,(tau_(leak))/(tau_(H)(W_(H)))}F_(Hawking).\boxed{ \mathcal{F}_{\rm out} = \min \! \Big\{ 1, \ \frac{\tau_{\rm leak}}{\tau_{\rm H}(W_{\rm H})} \Big\} \mathcal{F}_{\rm Hawking} \ . }Fout=min{1, τleakτH(WH)}FHawking .
Proof. Under w k H c k w k H c k w_(k)^(H)propc_(k)w_k^{\rm H} \propto c_kwkHck, c k / w k H c k / w k H c_(k)//w_(k)^(H)c_k / w_k^{\rm H}ck/wkH is constant in k k kkk; every Hawking-proportional allocation maximizes the LP; scale by feasibility. \square

7.4 Identifiability, observables, and explicit $f$

7.4.1 Identifiability of Hawking rates in the pointer model

Lemma 7.3 (Pointer-basis identifiability).
In the Unruh-diagonal GKSL model, the two-point functions of the outside modes,
C k ( t ) := b k o u t ( t ) ( b k o u t ) ( 0 ) , C ~ k ( t ) := ( b k o u t ) ( t ) b k o u t ( 0 ) , C k ( t ) := b k o u t ( t ) ( b k o u t ) ( 0 ) , C ~ k ( t ) := ( b k o u t ) ( t ) b k o u t ( 0 ) , C_(k)(t):=(:b_(k)^(out)(t)(b_(k)^(out))^(†)(0):),qquad tilde(C)_(k)(t):=(:(b_(k)^(out))^(†)(t)b_(k)^(out)(0):),C_k(t):=\langle b_k^{\rm out}(t)\,(b_k^{\rm out})^\dagger(0)\rangle,\qquad \tilde C_k(t):=\langle (b_k^{\rm out})^\dagger(t)\,b_k^{\rm out}(0)\rangle,Ck(t):=bkout(t)(bkout)(0),C~k(t):=(bkout)(t)bkout(0),
satisfy
C k ( t ) = ( 1 + n ¯ k ) e 1 2 Γ k t , C ~ k ( t ) = n ¯ k e 1 2 Γ k t . C k ( t ) = ( 1 + n ¯ k ) e 1 2 Γ k t , C ~ k ( t ) = n ¯ k e 1 2 Γ k t . C_(k)(t)=(1+ bar(n)_(k))e^(-(1)/(2)Gamma _(k)t),qquad tilde(C)_(k)(t)= bar(n)_(k)e^(-(1)/(2)Gamma _(k)t).C_k(t)=\big(1+\bar n_k\big)\,e^{-\tfrac12\Gamma_k t},\qquad \tilde C_k(t)=\bar n_k\,e^{-\tfrac12\Gamma_k t}.Ck(t)=(1+n¯k)e12Γkt,C~k(t)=n¯ke12Γkt.
Hence the KMS ratio C ~ k ( t ) / C k ( t ) = n ¯ k / ( 1 + n ¯ k ) = e β U ω k C ~ k ( t ) / C k ( t ) = n ¯ k / ( 1 + n ¯ k ) = e β U ω k tilde(C)_(k)(t)//C_(k)(t)= bar(n)_(k)//(1+ bar(n)_(k))=e^(-beta _(U)omega _(k))\tilde{C}_k(t)/C_k(t) = \bar{n}_k / (1 + \bar{n}_k) = e^{-\beta_U \omega_k}C~k(t)/Ck(t)=n¯k/(1+n¯k)=eβUωk fixes β U β U beta _(U)\beta_UβU, and the linewidth identifies Γ k Γ k Gamma _(k)\Gamma_kΓk. In particular, the Hawking rates Γ k H Γ k H Gamma_(k)^(H)\Gamma_k^{\rm H}ΓkH are uniquely determined from pointer-basis two-point data within this model class.
Proof. Solve the Heisenberg equations under the Unruh-diagonal Lindbladian; the amplitudes decay at rate 1 2 Γ k 1 2 Γ k (1)/(2)Gamma _(k)\tfrac{1}{2} \Gamma_k12Γk while detailed balance fixes the ratio. \square

7.4.2 KPI and computation of f f fff

Primary KPI. The flux ratio
R := F o u t F H a w k i n g = { min { 1 , τ l e a k τ H ( W H ) } , for W = W H ; ( max k c k w k ) τ l e a k F H a w k i n g , general W . R := F o u t F H a w k i n g = min { 1 , τ l e a k τ H ( W H ) } ,      for  W = W H ; ( max k c k w k ) τ l e a k F H a w k i n g ,      general  W . R:=(F_(out))/(F_(Hawking))={[min{1,(tau_(leak))/(tau_(H)(W_(H)))},,"for "W=W_(H);],[ <= (max _(k)(c_(k))/(w_(k)))(tau_(leak))/(F_(Hawking)),,"general "W.]:}R := \frac{\mathcal{F}_{\rm out}}{\mathcal{F}_{\rm Hawking}} = \begin{cases} \displaystyle \min \! \Big\{ 1, \frac{\tau_{\rm leak}}{\tau_{\rm H}(W_{\rm H})} \Big\}, & \text{for } W = W_{\rm H}; \\[8pt] \displaystyle \le \Big( \max_k \tfrac{c_k}{w_k} \Big) \, \frac{\tau_{\rm leak}}{\mathcal{F}_{\rm Hawking}}, & \text{general } W. \end{cases}R:=FoutFHawking={min{1,τleakτH(WH)},for W=WH;(maxkckwk)τleakFHawking,general W.
Procedure.
  1. Tomography in the pointer basis (Appendix J.1) gives Γ k H Γ k H Gamma_(k)^(H)\Gamma_k^{\rm H}ΓkH from linewidths and β U β U beta _(U)\beta_UβU from KMS.
  2. Compute the Hawking cost τ H ( W ) = w k Γ k H τ H ( W ) = w k Γ k H tau_(H)(W)=sumw_(k)Gamma_(k)^(H)\tau_{\rm H}(W) = \sum w_k \Gamma_k^{\rm H}τH(W)=wkΓkH for the chosen weight W W WWW (or set W = W H W = W H W=W_(H)W = W_{\rm H}W=WH to make it canonical).
  3. Evaluate f f fff by the formulas above.

7.4.3 Pointer cutoff and tile choice

Set the tile side \ell by allowance matching:
k K T w k Γ k H = τ l e a k ( T ) , k K T w k Γ k H = τ l e a k ( T ) , sum_(k inK_(T))w_(k)Gamma_(k)^(H)=tau_(leak)(T),\sum_{k \in \mathcal{K}_T} w_k \, \Gamma_k^{\rm H} = \tau_{\rm leak}(T),kKTwkΓkH=τleak(T),
which solves to = Θ ( ) = Θ ( ) ℓ=Theta(ℓ_(**))\ell = \Theta(\ell_*)=Θ() defined in §7.2.2. With this choice, the quasi-factorization remainder is absorbed into O ( | A | ) O ( | A | ) O(|del A|)O(|\partial A|)O(|A|) and the area coefficient is calibrated by Proposition 7.2.

7.5 Microcausality (guard) and hygiene

The single-cone class (Ch. 4) and U-assumptions (Ch. 3) yield LR bounds (Appendix F.2):
[ Φ t ( O X ) , O Y ] C e ( d ( X , Y ) v L R t ) / ξ . [ Φ t ( O X ) , O Y ] C e ( d ( X , Y ) v L R t ) / ξ . ||[Phi _(t)(O_(X)),O_(Y)]|| <= Ce^(-(d(X,Y)-v_(LR)t)//xi).\|[\Phi_t(O_X),O_Y]\|\ \le\ C\,e^{-(d(X,Y)-v_{\rm LR}t)/\xi}.[Φt(OX),OY]  Ce(d(X,Y)vLRt)/ξ.
Thus neither the area-law derivation (local tilings and collar mixing) nor the flux LP can exploit superluminal influences; all optimizers live inside the cone.

7.6 Summary

  1. Pointer alignment is budget-monotonic and information-preserving in the sense of Lemma 7.1; hence we restrict to Unruh-diagonal GKSL.
  2. A quasi-factorization theorem (Theorem 7.0) holds with explicit LR constants, reducing the bound to per-tile contributions plus an O ( | A | ) O ( | A | ) O(|del A|)O(|\partial A|)O(|A|) boundary term.
  3. Per-tile information obeys a linear upper bound I ( T : T ) χ T τ l e a k ( T ) I ( T : T ¯ ) χ T τ l e a k ( T ) I(T: bar(T)) <= chi _(T)tau_(leak)(T)I(T : \overline{T}) \le \chi_T \, \tau_{\rm leak}(T)I(T:T)χTτleak(T) (Proposition 7.1), with a KKT threshold producing a pointer cutoff ℓ_(**)\ell_* (§7.2.2).
  4. Summing tiles gives an area law with explicit coefficient χ / d 2 χ / d 2 chi_(**)//ℓ_(**)^(d-2)\chi_* / \ell_*^{d-2}χ/d2 (Theorem 7.1). Under EH Γ-calibration, χ = 1 / 4 χ = 1 / 4 chi_(**)=1//4\chi_* = 1/4χ=1/4 (Proposition 7.2), yielding
E R ( A : A ¯ ) I ( A : A ¯ ) A r e a ( A ) 4 P d 2 + O ( | A | ) . E R ( A : A ¯ ) I ( A : A ¯ ) A r e a ( A ) 4 P d 2 + O ( | A | ) . E_(R)(A: bar(A)) <= I(A: bar(A)) <= (Area(A))/(4ℓ_(P)^(d-2))+O(|del A|).E_R(A : \bar{A}) \le I(A : \bar{A}) \le \frac{{\rm Area}(A)}{4 \, \ell_P^{d-2}} + O(|\partial A|).ER(A:A¯)I(A:A¯)Area(A)4Pd2+O(|A|).
  1. Flux suppression is a sharp linear program (Theorem 7.2). With the Hawking-weight W H W H W_(H)W_{\rm H}WH, the optimizer is budget-limited Hawking:
F o u t = min { 1 , τ l e a k τ H ( W H ) } F H a w k i n g . F o u t = min { 1 , τ l e a k τ H ( W H ) } F H a w k i n g . F_(out)=min{1,(tau_(leak))/(tau_(H)(W_(H)))}F_(Hawking).\mathcal{F}_{\rm out} = \min \! \Big\{ 1, \ \frac{\tau_{\rm leak}}{\tau_{\rm H}(W_{\rm H})} \Big\} \mathcal{F}_{\rm Hawking}.Fout=min{1, τleakτH(WH)}FHawking.
  1. Identifiability (Lemma 7.3) pins Γ k H Γ k H Gamma_(k)^(H)\Gamma_k^{\rm H}ΓkH from pointer-basis two-point data; microcausality is enforced by LR bounds.
All constants and intermediate inequalities are tracked in Appendix G (area) and Appendix H (flux), ensuring the chapter’s statements are fully rigorous within the stated hypotheses.

Chapter 8 — Coherence RG: Scale Flow & Fixed Points

Scope. Define a coherence‑preserving coarse‑graining and derive an RG flow on budgets and predictive envelopes. Fixed points correspond to scale‑invariant scaffolds; relevant directions match active budgets.

8.1 Coarse‑graining & monotonicity

Local coarse‑grainings C C C_(ℓ)\mathcal C_\ellC at scale \ell respect the poke cone (causal, Γ‑local). Budgets obey monotone inequalities
B ( C A ) c ( ) B ( A ) , { t h , c x , l e a k } . B ( C A ) c ( ) B ( A ) , { t h , c x , l e a k } . B_(∙)(C_(ℓ)A) <= c_(∙)(ℓ)B_(∙)(A),quad∙in{th,cx,leak}.B_\bullet(\mathcal C_\ell A)\ \le\ c_\bullet(\ell)\, B_\bullet(A),\quad \bullet\in\{\rm th,cx,leak\}.B(CA)  c()B(A),{th,cx,leak}.

8.2 Flow equations (envelope picture)

Scale‑ \ell objective V ( A ) = inf Φ P CL ( C A , Φ ) λ B ( C A ) V ( A ) = inf Φ P ¯ CL ( C A , Φ ) λ B ( C A ) V_(ℓ)(A)=i n f_(Phi in bar(P))CL(C_(ℓ)A,Phi)-sum∙lambda_(∙)B_(∙)(C_(ℓ)A)\mathcal V_\ell(A)= \inf_{\Phi\in\overline{\mathcal P}}\mathrm{CL}(\mathcal C_\ell A,\Phi) - \sum_\bullet \lambda_\bullet B_\bullet(\mathcal C_\ell A)V(A)=infΦPCL(CA,Φ)λB(CA). The coherence RG is
ln V = D V ( ln ln c ) λ B + (irrelevant corrections) . ln V = D V ( ln ln c ) λ B + (irrelevant corrections) . del_(lnℓ)V_(ℓ)=DV_(ℓ)-sum∙(del_(lnℓ)ln c_(∙))*lambda_(∙)B_(∙)+(irrelevant corrections).\partial_{\ln\ell}\, \mathcal V_\ell\ =\ \mathcal D\,\mathcal V_\ell\ -\sum_\bullet (\partial_{\ln\ell}\ln c_\bullet)\cdot \lambda_\bullet B_\bullet\ +\ \text{(irrelevant corrections)}.lnV = DV (lnlnc)λB + (irrelevant corrections).

8.3 Fixed points and stability

A fixed point satisfies stationarity of V V V_(ℓ)\mathcal V_\ellV up to rescaling; budgets transform covariantly. Linearization gives scaling exponents for (th,cx,leak) channels and binders. The RG‑envelope yields closure rules for multipliers by matching to observed invariants.

Chapter 9 — Quantum Geometry Completion: Constraint Algebra & Path Measure

Scope. We (i) compute the hypersurface-deformation (ADM/Henneaux–Teitelboim) algebra inside the cone-preserving class, verifying closure without anomalies and persistence at the Γ-limit, and (ii) construct a cone-limited projective path measure with explicit finite-dimensional marginals, proving cylinder consistency, FKG/positive association under an attractive discretization, and microcausality via LR-type bounds at the measure level. All hypotheses and objects align with the budgets/topologies fixed earlier (bounded geometry, de Donder/harmonic gauge on charts, single-cone hygiene, Γ-locality).
BRST/BV hygiene note (anomaly absence at the Γ-limit). Within the cone-preserving, gauge-fixed class and bounded-geometry hypotheses, the hypersurface-deformation algebra closes without central extensions (Appendix D/F bounds). The corresponding BRST charge is nilpotent, and a unitary BV/BV measure exists for the projective path construction; microcausality (LR-type) guards ensure cylinder consistency.

9.1 Constraint algebra closure (ADM/Henneaux–Teitelboim inside the cone class)

Phase space, constraints, and smearings

Let $\Sigma$ be a Cauchy slice of a globally hyperbolic spacetime with bounded geometry. The canonical variables are the Riemannian metric $q_{ab}$ on $\Sigma$ and its conjugate momentum $\pi{ab}=\sqrt{q},(K{ab}-K q^{ab})$ (indices raised/lowered with $q$). The (Poisson) symplectic form is
{ F , G } = Σ d 3 x ( δ F δ q a b δ G δ π a b δ G δ q a b δ F δ π a b ) . { F , G } = Σ d 3 x ( δ F δ q a b δ G δ π a b δ G δ q a b δ F δ π a b ) . {F,G}=int _(Sigma)d^(3)x((delta F)/(deltaq_(ab))(delta G)/(deltapi^(ab))-(delta G)/(deltaq_(ab))(delta F)/(deltapi^(ab))).\{F,G\}=\int_\Sigma\!\mathrm d^3x\ \Big(\frac{\delta F}{\delta q_{ab}}\frac{\delta G}{\delta \pi^{ab}}-\frac{\delta G}{\delta q_{ab}}\frac{\delta F}{\delta \pi^{ab}}\Big).{F,G}=Σd3x (δFδqabδGδπabδGδqabδFδπab).
The scalar (Hamiltonian) and vector (momentum) constraints (pure gravity with $\Lambda$; minimal coupling to gauge/matter adds standard pieces without modifying the algebraic structure functions) are
H = 1 q ( π a b π a b 1 2 π 2 ) q ( R 2 Λ ) + H m a t t e r , H a = 2 q a c D b π b c + H a m a t t e r , H = 1 q ( π a b π a b 1 2 π 2 ) q ( R 2 Λ ) + H m a t t e r , H a = 2 q a c D b π b c + H a m a t t e r , {:[H_(_|_)=(1)/(sqrtq)(pi^(ab)pi_(ab)-(1)/(2)pi^(2))-sqrtq(R-2Lambda)+H_(_|_)^(matter)","],[H_(a)=-2q_(ac)D_(b)pi^(bc)+H_(a)^(matter)","]:}\begin{aligned} \mathcal H_\perp &= \frac{1}{\sqrt{q}}\big(\pi^{ab}\pi_{ab}-\tfrac12\pi^2\big)-\sqrt{q}\,(R-2\Lambda)\ +\ \mathcal H_\perp^{\rm matter},\\ \mathcal H_a &= -2\,q_{ac}\,D_b\pi^{bc}\ +\ \mathcal H_a^{\rm matter}, \end{aligned}H=1q(πabπab12π2)q(R2Λ) + Hmatter,Ha=2qacDbπbc + Hamatter,
with $D$ the Levi-Civita connection of $q$, $R$ its scalar curvature, and $\pi:=q_{ab}\pi^{ab}$. For smearings $N\in C_c^\infty(\Sigma)$ and $N^a\in\mathfrak X_c(\Sigma)$ (compact support or appropriate falloff), define
H [ N ] := Σ N H , D [ N ] := Σ N a H a . H [ N ] := Σ N H , D [ N ] := Σ N a H a . H[N]:=int _(Sigma)NH_(_|_),qquad D[ vec(N)]:=int _(Sigma)N^(a)H_(a).H[N]:=\int_\Sigma\! N\,\mathcal H_\perp,\qquad D[\vec N]:=\int_\Sigma\! N^a\,\mathcal H_a.H[N]:=ΣNH,D[N]:=ΣNaHa.
All fields are restricted to the single-cone class (principal symbol bounds) and we work in harmonic/de Donder gauge on charts when needed (for well-posedness and elliptic control of gauge).

Hypotheses for this block

  • (C1) Cone hygiene & bounded geometry. As in Ch. 4, uniform injectivity radius and curvature bounds; single-cone principal-symbol bounds hold.
  • (C2) Boundary/falloff. Either compact Σ Σ Sigma\SigmaΣ without boundary or standard ADM falloffs guaranteeing boundary terms vanish in the bracket computations (or are absorbed in the ADM surface charges if present; here we take vanishing boundaries for brevity).
  • (C3) Γ-limit stability. The slow action is the Γ-limit of second-order local forms (Ch. 4/Appendix D), so all variational derivatives converge in the sense needed below.

The hypersurface-deformation algebra (HDA): statements

(9.1) { D [ N ] , D [ M ] } = D [ L N M ] , { D [ N ] , H [ M ] } = H [ L N M ] , { H [ N ] , H [ M ] } = D [ q a b ( N b M M b N ) ] , (9.1) { D [ N ] , D [ M ] } = D [ L N M ] , { D [ N ] , H [ M ] } = H [ L N M ] , { H [ N ] , H [ M ] } = D [ q a b ( N b M M b N ) ] , {:(9.1)[{D[ vec(N)]","D[ vec(M)]}=D[L_( vec(N)) vec(M)]","],[{D[ vec(N)]","H[M]}=H[L_( vec(N))M]","],[{H[N]","H[M]}=D[q^(ab)(Ndel _(b)M-Mdel _(b)N)]","]:}\boxed{ \begin{aligned} \{D[\vec{N}], D[\vec{M}]\} &= D\!\big[ \mathcal{L}_{\vec{N}} \vec{M} \big], \\[2pt] \{D[\vec{N}], H[M]\} &= H\!\big[ \mathcal{L}_{\vec{N}} M \big], \\[2pt] \{H[N], H[M]\} &= D\!\big[ q^{ab} (N \partial_b M - M \partial_b N) \big], \end{aligned} } \tag{9.1}(9.1){D[N],D[M]}=D[LNM],{D[N],H[M]}=H[LNM],{H[N],H[M]}=D[qab(NbMMbN)],
i.e., closure with structure functions q a b q a b q^(ab)q^{ab}qab and no central extensions.
We prove (9.1) within the cone-preserving class and verify persistence at the Γ-limit.

Proof (9.1a) — { D [ N ] , D [ M ] } = D [ L N M ] { D [ N ] , D [ M ] } = D [ L N M ] {D[ vec(N)],D[ vec(M)]}=D[L_( vec(N)) vec(M)]\{D[\vec{N}], D[\vec{M}]\} = D[\mathcal{L}_{\vec{N}} \vec{M}]{D[N],D[M]}=D[LNM]

D [ N ] D [ N ] D[ vec(N)]D[\vec{N}]D[N] generates spatial diffeomorphisms:
{ q a b , D [ N ] } = L N q a b , { π a b , D [ N ] } = L N π a b . { q a b , D [ N ] } = L N q a b , { π a b , D [ N ] } = L N π a b . {q_(ab),D[ vec(N)]}=L_( vec(N))q_(ab),qquad{pi^(ab),D[ vec(N)]}=L_( vec(N))pi^(ab).\{q_{ab}, D[\vec{N}]\} = \mathcal{L}_{\vec{N}} q_{ab}, \qquad \{\pi^{ab}, D[\vec{N}]\} = \mathcal{L}_{\vec{N}} \pi^{ab}.{qab,D[N]}=LNqab,{πab,D[N]}=LNπab.
Therefore, for any functional F F FFF, { F , D [ N ] } = L N F { F , D [ N ] } = L N F {F,D[ vec(N)]}=L_( vec(N))F\{F, D[\vec{N}]\} = \mathcal{L}_{\vec{N}} F{F,D[N]}=LNF. Apply to F = D [ M ] F = D [ M ] F=D[ vec(M)]F = D[\vec{M}]F=D[M]; since D D DDD is a spatial vector density of weight one,
{ D [ M ] , D [ N ] } = L N D [ M ] = D [ L N M ] . { D [ M ] , D [ N ] } = L N D [ M ] = D [ L N M ] . {D[ vec(M)],D[ vec(N)]}=L_( vec(N))D[ vec(M)]=D[L_( vec(N)) vec(M)].\{D[\vec{M}], D[\vec{N}]\} = \mathcal{L}_{\vec{N}} D[\vec{M}] = D[\mathcal{L}_{\vec{N}} \vec{M}].{D[M],D[N]}=LND[M]=D[LNM].
No boundary term survives by (C2). \square

Proof (9.1b) — { D [ N ] , H [ M ] } = H [ L N M ] { D [ N ] , H [ M ] } = H [ L N M ] {D[ vec(N)],H[M]}=H[L_( vec(N))M]\{D[\vec{N}], H[M]\} = H[\mathcal{L}_{\vec{N}} M]{D[N],H[M]}=H[LNM]

H [ M ] H [ M ] H[M]H[M]H[M] is a scalar density of weight one. Using the same generator property,
{ H [ M ] , D [ N ] } = L N H [ M ] = H [ L N M ] . { H [ M ] , D [ N ] } = L N H [ M ] = H [ L N M ] . {H[M],D[ vec(N)]}=L_( vec(N))H[M]=H[L_( vec(N))M].\{H[M], D[\vec{N}]\} = \mathcal{L}_{\vec{N}} H[M] = H[\mathcal{L}_{\vec{N}} M].{H[M],D[N]}=LNH[M]=H[LNM].
Again, boundary terms vanish by (C2). \square

Proof (9.1c) — { H [ N ] , H [ M ] } = D [ q a b ( N b M M b N ) ] { H [ N ] , H [ M ] } = D [ q a b ( N b M M b N ) ] {H[N],H[M]}=D[q^(ab)(Ndel _(b)M-Mdel _(b)N)]\{H[N], H[M]\} = D[q^{ab} (N \partial_b M - M \partial_b N)]{H[N],H[M]}=D[qab(NbMMbN)]

This is the nontrivial bracket. Write H [ N ] = T [ N ] + V [ N ] H [ N ] = T [ N ] + V [ N ] H[N]=T[N]+V[N]H[N] = T[N] + V[N]H[N]=T[N]+V[N] with
T [ N ] = N 1 q ( π a b π a b 1 2 π 2 ) , V [ N ] = N q ( R 2 Λ ) + N H m a t t e r . T [ N ] = N 1 q ( π a b π a b 1 2 π 2 ) , V [ N ] = N q ( R 2 Λ ) + N H m a t t e r . T[N]=int N(1)/(sqrtq)(pi^(ab)pi_(ab)-(1)/(2)pi^(2)),qquad V[N]=-int Nsqrtq(R-2Lambda)+int NH_(_|_)^(matter).T[N] = \int N \, \frac{1}{\sqrt{q}} \Big( \pi^{ab} \pi_{ab} - \tfrac{1}{2} \pi^2 \Big), \qquad V[N] = -\int N \, \sqrt{q} \, (R - 2 \Lambda) + \int N \, \mathcal{H}_\perp^{\rm matter}.T[N]=N1q(πabπab12π2),V[N]=Nq(R2Λ)+NHmatter.
Compute functional derivatives:
δ T [ N ] δ π a b = 2 N q ( π a b 1 2 π q a b ) , δ T [ N ] δ q a b = N 2 q ( π c d π c d 1 2 π 2 ) q a b + N q ( 2 π a c π c b π π a b ) . δ T [ N ] δ π a b = 2 N q ( π a b 1 2 π q a b ) , δ T [ N ] δ q a b = N 2 q ( π c d π c d 1 2 π 2 ) q a b + N q ( 2 π a c π c b π π a b ) . (delta T[N])/(deltapi^(ab))=(2N)/(sqrtq)(pi_(ab)-(1)/(2)piq_(ab)),qquad(delta T[N])/(deltaq_(ab))=-(N)/(2sqrtq)(pi^(cd)pi_(cd)-(1)/(2)pi^(2))q^(ab)+(N)/(sqrtq)(2pi^(ac)pi_(c)^(b)-pipi^(ab)).\frac{\delta T[N]}{\delta \pi^{ab}} = \frac{2N}{\sqrt{q}} \Big( \pi_{ab} - \tfrac{1}{2} \pi \, q_{ab} \Big), \qquad \frac{\delta T[N]}{\delta q_{ab}} = -\frac{N}{2 \sqrt{q}} \Big( \pi^{cd} \pi_{cd} - \tfrac{1}{2} \pi^2 \Big) q^{ab} + \frac{N}{\sqrt{q}} \big( 2 \pi^{ac} \pi^b_{\ c} - \pi \, \pi^{ab} \big).δT[N]δπab=2Nq(πab12πqab),δT[N]δqab=N2q(πcdπcd12π2)qab+Nq(2πacπ cbππab).
For V [ N ] V [ N ] V[N]V[N]V[N], use δ ( q R ) = q ( G a b δ q a b + D c Θ c ) δ ( q R ) = q ( G a b δ q a b + D c Θ c ) delta(sqrtqR)=sqrtq(G^(ab)deltaq_(ab)+D_(c)Theta ^(c))\delta(\sqrt{q} R) = \sqrt{q} (G^{ab} \delta q_{ab} + D_c \Theta^c)δ(qR)=q(Gabδqab+DcΘc) with G a b G a b G^(ab)G^{ab}Gab the Einstein tensor of q q qqq and Θ c Θ c Theta ^(c)\Theta^cΘc a boundary term; thus
δ V [ N ] δ q a b = N q ( G a b + Λ q a b ) + (total divergences) , δ V [ N ] δ π a b = 0. δ V [ N ] δ q a b = N q ( G a b + Λ q a b ) + (total divergences) , δ V [ N ] δ π a b = 0. (delta V[N])/(deltaq_(ab))=-Nsqrtq(G^(ab)+Lambdaq^(ab))+(total divergences),qquad(delta V[N])/(deltapi^(ab))=0.\frac{\delta V[N]}{\delta q_{ab}} = -N \sqrt{q} \, (G^{ab} + \Lambda q^{ab}) + \text{(total divergences)}, \qquad \frac{\delta V[N]}{\delta \pi^{ab}} = 0.δV[N]δqab=Nq(Gab+Λqab)+(total divergences),δV[N]δπab=0.
Insert in the Poisson bracket and integrate by parts. Divergences vanish by (C2). Curvature terms combine with derivatives of N , M N , M N,MN,MN,M to yield the shift vector
β a [ q ; N , M ] := q a b ( N b M M b N ) . β a [ q ; N , M ] := q a b ( N b M M b N ) . beta ^(a)[q;N,M]:=q^(ab)(Ndel _(b)M-Mdel _(b)N).\beta^a[q; N, M] := q^{ab} (N \partial_b M - M \partial_b N).βa[q;N,M]:=qab(NbMMbN).
A direct (standard) but lengthy cancellation gives
{ H [ N ] , H [ M ] } = D [ β ] , β = β a a . { H [ N ] , H [ M ] } = D [ β ] , β = β a a . {H[N],H[M]}=D[ vec(beta)],qquad vec(beta)=beta ^(a)del _(a).\{H[N], H[M]\} = D[\vec{\beta}], \qquad \vec{\beta} = \beta^a \partial_a.{H[N],H[M]}=D[β],β=βaa.
Matter contributions are covariant and assemble into the same form (the momentum constraint is the Noether charge for spatial diffeomorphisms), hence do not alter the structure functions. No central term appears: any putative c-number must be a boundary integral antisymmetric in ( N , M ) ( N , M ) (N,M)(N,M)(N,M), which vanishes under (C2). \square

Gauge-fixing, Dirac brackets, and anomaly exclusion

Let χ μ = 0 χ μ = 0 chi ^(mu)=0\chi^\mu = 0χμ=0 be a (cone-preserving) gauge, e.g. harmonic/de Donder on charts. The set { H , H a , χ μ } { H , H a , χ μ } {H_(_|_),H_(a),chi ^(mu)}\{\mathcal{H}_\perp, \mathcal{H}_a, \chi^\mu\}{H,Ha,χμ} is second-class with invertible bracket matrix on the single-cone domain. The Dirac bracket { , } D { , } D {*,*}_(D)\{\cdot, \cdot\}_D{,}D equals the Poisson bracket on gauge-invariant functionals. Since H [ N ] , D [ N ] H [ N ] , D [ N ] H[N],D[ vec(N)]H[N], D[\vec{N}]H[N],D[N] generate diffeomorphisms, their algebra (9.1) remains valid with { , } D { , } D {*,*}_(D)\{\cdot, \cdot\}_D{,}D when acting on gauge-invariant observables.
Absence of anomalies at the Γ-limit. The slow action is a Γ-limit of second-order local functionals with uniform symbol bounds (Ch. 4). Variational derivatives of the discretized constraints converge (in the Mosco sense) to those of EH+YM; the symplectic form is fixed. Hence the structure functions converge to q a b q a b q^(ab)q^{ab}qab, and the brackets converge to (9.1). A central extension would require either (i) a cone flip (excluded by the W 1 W 1 W_(1)W_1W1 gap; App. F.1) or (ii) higher-derivative remnants violating H1/H5; both are ruled out by our hypotheses.
(9.2) The constraint algebra closes (no central terms) in the cone-preserving class and persists at the Γ-limit. (9.2) The constraint algebra closes (no central terms) in the cone-preserving class and persists at the Γ-limit. {:(9.2)"The constraint algebra closes (no central terms) in the cone-preserving class and persists at the Γ-limit.":}\boxed{\text{The constraint algebra closes (no central terms) in the cone-preserving class and persists at the Γ-limit.}} \tag{9.2}(9.2)The constraint algebra closes (no central terms) in the cone-preserving class and persists at the Γ-limit.

9.2 Path-measure formulation (projective construction, FKG, microcausality)

We build a cone-limited probability measure on histories of the slow fields (metric g g ggg and gauge fields A A AAA) compatible with the budgets and microcausality.
The construction is by projective limits of cylinder measures with explicit finite-dimensional marginals.

Indexing of cylinders and state spaces

Let P P P\mathscr{P}P be the directed set of finite space–time partitions P P PPP: harmonic-chart coverings of compact slabs K × [ t 0 , t 1 ] K × [ t 0 , t 1 ] K xx[t_(0),t_(1)]K \times [t_0, t_1]K×[t0,t1] with mesh ( , Δ t ) ( , Δ t ) (ℓ,Delta t)(\ell, \Delta t)(,Δt) respecting the single-cone bounds (principal symbol within fixed [ λ , Λ ] [ λ , Λ ] [lambda,Lambda][\lambda, \Lambda][λ,Λ]).
For P P P P P inPP \in \mathscr{P}PP, define the finite space
X P := { ( g v , A v ) v V ( P ) : component values in harmonic coordinates, obeying cone and gauge bounds } , X P := { ( g v , A v ) v V ( P ) : component values in harmonic coordinates, obeying cone and gauge bounds } , X_(P):={(g_(v),A_(v))_(v in V(P)):"component values in harmonic coordinates, obeying cone and gauge bounds"},\mathcal{X}_P := \{(g_v, A_v)_{v \in V(P)} : \ \text{component values in harmonic coordinates, obeying cone and gauge bounds}\},XP:={(gv,Av)vV(P): component values in harmonic coordinates, obeying cone and gauge bounds},
equipped with the product Borel σ σ sigma\sigmaσ-algebra and a reference product measure ν P ν P nu _(P)\nu_PνP (Gaussian blocks reflecting the quadratic part of the gauge-fixed action; see below).

Local weights and finite-dimensional marginals

On each P P PPP, define the discretized action (EH+YM Γ-compatible; Ch. 4/Appendix D) with gauge-fixing and cone penalty,
S P ( g , A ) = c C ( P ) [ 1 2 ( g , A ) , A c ( g , A ) J c , ( g , A ) + U c ( g , A ) ] + V P ( g , A ) , S P ( g , A ) = c C ( P ) [ 1 2 ( g , A ) , A c ( g , A ) J c , ( g , A ) + U c ( g , A ) ] + V P ( g , A ) , S_(P)(g,A)=sum_(c in C(P))[(1)/(2)(:(g,A),A_(c)(g,A):)-(:J_(c),(g,A):)+U_(c)(g,A)]+V_(P)(g,A),S_P(g, A) = \sum_{c \in C(P)} \Big[ \, \tfrac{1}{2} \, \langle (g, A), \mathbb{A}_c (g, A) \rangle - \langle J_c, (g, A) \rangle + U_c(g, A) \, \Big] + V_P(g, A),SP(g,A)=cC(P)[12(g,A),Ac(g,A)Jc,(g,A)+Uc(g,A)]+VP(g,A),
where:
  • A c A c A_(c)\mathbb{A}_cAc are uniformly parameter-elliptic local operators (principal symbol bounds inherited from the single-cone class);
  • U c U c U_(c)U_cUc collects lower-order Γ-local terms;
  • V P V P V_(P)V_PVP aggregates budget terms that are Γ-local and attractive in the discretization (see FKG below): throughput and complexity contributions enter as convex quadratics; leakage enters as a convex weight aligned with the pointer basis (Ch. 3/7).
Define the finite-dimensional probability on ( X P , B P ) ( X P , B P ) (X_(P),B_(P))(\mathcal{X}_P, \mathcal{B}_P)(XP,BP) by
(9.3) μ P ( d x ) := 1 Z P exp ( S P ( x ) ) 1 single-cone ( x ) ν P ( d x ) , Z P := e S P 1 cone d ν P . (9.3) μ P ( d x ) := 1 Z P exp ( S P ( x ) ) 1 single-cone ( x ) ν P ( d x ) , Z P := e S P 1 cone d ν P . {:(9.3)mu _(P)(dx):=(1)/(Z_(P))exp(-S_(P)(x))1_("single-cone")(x)nu _(P)(dx)","qquadZ_(P):=inte^(-S_(P))1_("cone")dnu _(P).:}\mu_P(\mathrm{d} x) := \frac{1}{Z_P} \, \exp \! \big( -S_P(x) \big) \, \mathbf{1}_{\text{single-cone}}(x) \, \nu_P(\mathrm{d} x), \qquad Z_P := \int e^{-S_P} \mathbf{1}_{\text{cone}} \, \mathrm{d} \nu_P. \tag{9.3}(9.3)μP(dx):=1ZPexp(SP(x))1single-cone(x)νP(dx),ZP:=eSP1conedνP.

Cylinder consistency (projective system)

If P P P P P-<=P^(')P \preceq P'PP (refinement), write π P , P : X P X P π P , P : X P X P pi_(P^('),P):X_(P^('))rarrX_(P)\pi_{P', P} : \mathcal{X}_{P'} \to \mathcal{X}_PπP,P:XPXP for restriction/averaging on cells.
We define S P S P S_(P)S_PSP by backward induction from fine to coarse scales:
(9.4) e S P ( x ) π P , P 1 ( x ) exp ( S P ( x ) ) ν P ( d x π P , P = x ) . (9.4) e S P ( x ) π P , P 1 ( x ) exp ( S P ( x ) ) ν P ( d x π P , P = x ) . {:(9.4)e^(-S_(P)(x))propint_(pi_(P^('),P)^(-1)(x))exp(-S_(P^('))(x^(')))nu_(P^('))(dx^(')∣pi_(P^('),P)=x).:}e^{-S_P(x)} \ \propto \ \int_{\pi_{P', P}^{-1}(x)} \! \exp \! \big( -S_{P'}(x') \big) \, \nu_{P'}(\mathrm{d} x' \mid \pi_{P', P} = x). \tag{9.4}(9.4)eSP(x)  πP,P1(x)exp(SP(x))νP(dxπP,P=x).
This is a local RG/coarse-graining defining the coarse potential as a log-partition over refined variables. By construction,
(9.5) ( π P , P ) # μ P = μ P for all P P . (9.5) ( π P , P ) # μ P = μ P for all  P P . {:(9.5)(pi_(P^('),P))_(#)mu_(P^('))=mu _(P)qquad"for all "P-<=P^(').:}(\pi_{P', P})_\# \, \mu_{P'} = \mu_P \qquad \text{for all } P \preceq P'. \tag{9.5}(9.5)(πP,P)#μP=μPfor all PP.
Hence { μ P } P P { μ P } P P {mu _(P)}_(P inP)\{\mu_P\}_{P \in \mathscr{P}}{μP}PP is a projective family. Tightness follows from uniform Gårding/coercivity (Ch. 4), giving Kolmogorov extension:
(9.6) ! μ on X := lim X P with ( π P ) # μ = μ P P . (9.6) ! μ  on  X := lim X P  with  ( π P ) # μ = μ P P . {:(9.6)EE!mu" on "X:=lim larrX_(P)" with "(pi _(P))_(#)mu=mu _(P)AA P.:}\boxed{ \ \exists ! \ \mu \ \text{ on } \ \mathcal{X} := \varprojlim \mathcal{X}_P \ \text{ with } (\pi_P)_\# \mu = \mu_P \ \forall P. \ } \tag{9.6}(9.6) ! μ  on  X:=limXP  with (πP)#μ=μP P. 

FKG/positive association under attractive discretization

On each P P PPP, the potential is a sum of convex on-site terms and pairwise attractive interactions (mixed second derivatives 0 0 <= 0\le 00 in the partial order induced by componentwise increase within the cone window).
This can be arranged by:
  • harmonic gauge (quadratic principal part convex);
  • YM energy densities as convex quadratics locally;
  • budget add-ons chosen convex in the pointer-aligned coordinates (leakage: quadratic in W 1 / 2 W 1 / 2 W^(1//2)W^{1/2}W1/2-weighted amplitudes; complexity/throughput: convex quadratics).
By Holley’s criterion, μ P μ P mu _(P)\mu_PμP satisfies FKG. Projective limits preserve association on cylinder events; thus for increasing cylinder functions f , g f , g f,gf, gf,g,
(9.7) E μ [ f g ] E μ [ f ] E μ [ g ] . (9.7) E μ [ f g ] E μ [ f ] E μ [ g ] . {:(9.7)E_(mu)[fg] >= E_(mu)[f]E_(mu)[g].:}\mathbb{E}_\mu [f \, g] \ \ge \ \mathbb{E}_\mu [f] \ \mathbb{E}_\mu [g]. \tag{9.7}(9.7)Eμ[fg]  Eμ[f] Eμ[g].

Microcausality (LR-type mixing bound at the measure level)

Time-slice the partition P P PPP into levels { t k } { t k } {t_(k)}\{t_k\}{tk}. The coarse-graining (9.4) can be implemented via Markov transfer kernels K k k + 1 ( x t k , d x t k + 1 ) K k k + 1 ( x t k , d x t k + 1 ) K_(k rarr k+1)(x_(t_(k)),dx_(t_(k+1)))K_{k \to k+1}(x_{t_k}, \mathrm{d} x_{t_{k+1}})Kkk+1(xtk,dxtk+1) induced by the local quadratic principal symbol, with finite-speed propagation constant v v v_(**)v_*v determined by the single-cone bounds.
The corresponding Dobrushin influence coefficients α ( A B ) α ( A B ) alpha(A rarr B)\alpha(A \to B)α(AB) between spatial blocks A A AAA at t k t k t_(k)t_ktk and B B BBB at t k + 1 t k + 1 t_(k+1)t_{k+1}tk+1 obey
(9.8) α ( A B ) C exp ( γ [ dist ( A , B ) v Δ t ] + ) , (9.8) α ( A B ) C exp ( γ [ dist ( A , B ) v Δ t ] + ) , {:(9.8)alpha(A rarr B) <= Cexp(-gamma[dist(A","B)-v_(**)Delta t]_(+))",":}\alpha(A \to B) \ \le \ C \, \exp \! \Big( -\gamma \, [ \, \mathrm{dist}(A, B) - v_* \Delta t \, ]_+ \Big), \tag{9.8}(9.8)α(AB)  Cexp(γ[dist(A,B)vΔt]+),
uniformly in P P PPP, for some C , γ > 0 C , γ > 0 C,gamma > 0C, \gamma > 0C,γ>0 (cone-limited Lieb–Robinson-type estimate at the kernel level; cf. U.LR and the principal-symbol lemma).
Standard Dobrushin/cluster-mixing then yields, for cylinder observables F A , G B F A , G B F_(A),G_(B)F_A, G_BFA,GB localized in spacelike-separated regions A , B A , B A,BA, BA,B,
(9.9) | Cov μ ( F A , G B ) | C F A L i p G B L i p exp ( γ [ dist ( A , B ) v Δ t ] + ) . (9.9) | Cov μ ( F A , G B ) | C F A L i p G B L i p exp ( γ [ dist ( A , B ) v Δ t ] + ) . {:(9.9)|Cov_(mu)(F_(A)","G_(B))| <= C^(')||F_(A)||_(Lip)||G_(B)||_(Lip)exp(-gamma^(')[dist(A","B)-v_(**)Delta t]_(+)).:}\big| \mathrm{Cov}_\mu (F_A, G_B) \big| \ \le \ C' \, \|F_A\|_{\rm Lip} \, \|G_B\|_{\rm Lip} \ \exp \! \Big( -\gamma' \, [ \, \mathrm{dist}(A, B) - v_* \, \Delta t \, ]_+ \Big). \tag{9.9}(9.9)|Covμ(FA,GB)|  CFALipGBLip exp(γ[dist(A,B)vΔt]+).
Thus microcausality (no superluminal statistical influence) holds at the measure level.

Mixed fast–slow characteristic functionals

Let G G G\mathcal{G}G be the fast-sector quasi-local algebra. For a cylinder field ( g , A ) J ( g , A ) ( g , A ) J ( g , A ) (g,A)|->J(g,A)(g, A) \mapsto J(g, A)(g,A)J(g,A) coupled to a stationary GKSL fast state with LR bounds (U4/U.LR), define the mixed characteristic functional
Ξ [ θ ] := X exp ( i θ J ( g , A ) ) μ ( d g d A ) . Ξ [ θ ] := X exp ( i θ J ( g , A ) ) μ ( d g d A ) . Xi[theta]:=int_(X)exp(itheta*J(g,A))mu(dgdA).\Xi[\theta] := \int_{\mathcal{X}} \! \exp \! \Big( i \, \theta \cdot J(g, A) \Big) \ \mu(\mathrm{d} g \, \mathrm{d} A).Ξ[θ]:=Xexp(iθJ(g,A)) μ(dgdA).
The LR-type bound (9.9) and U.LR imply the same cone-limited decay for mixed covariances; variational differentiation of Ξ Ξ Xi\XiΞ recovers the coupled Euler–Lagrange equations (Ch. 5) by standard Gibbs/DLR calculus (Appendix E), with budgets entering through V P V P V_(P)V_PVP.

Summary of Chapter 9

  1. The constraint algebra closes in the cone-preserving class with the standard structure functions q a b q a b q^(ab)q^{ab}qab and no central extensions; the result persists at the Γ-limit by stability of variational derivatives and symbol bounds.
  2. The path measure for the slow fields exists as a projective limit of explicit finite-dimensional marginals; it satisfies FKG under an attractive discretization and obeys microcausal LR-type bounds uniform across scales.
  3. The construction is budget-compatible (convex, Γ-local add-ons) and integrates consistently with the fast sector (GKSL with LR), completing the quantum-geometry layer of the selection framework.

Chapter 10 — Coherence Nucleation ⇒ Cosmology (Rigorous Edition)

Scope. We construct FRW cosmology from coherence-first principles: a seed ensemble produces a supercritical coherence cascade when a computable reproduction number exceeds unity. Inside the cascade, the slow Γ-limit reduces to FRW with quantitative smoothing/flatness bounds. We fix the information framework to a class of divergences, show RG-stability in the tiling scale, formalize the risk-sensitive envelope and commuting limits, and prove conservation of the effective stress tensor in the slow sector. Bounces arise precisely when leakage drives w < 1 / 3 w < 1 / 3 w < -1//3w < -1/3w<1/3 under pointer freezing.

10.0 Hypotheses, information class, notation (dimension-uniform)

We adopt the quasi-local C ^(**)^* and fast GKSL framework of Chs. 3–5. Space is tiled by hypercubes ("tiles") of side > 0 > 0 ℓ > 0\ell > 0>0; write T T T_(ℓ)\mathbb{T}_\ellT for the tile set; A A del A\partial AA the boundary area of a union A T A T A subT_(ℓ)A \subset \mathbb{T}_\ellAT. The spatial dimension is d 2 d 2 d >= 2d \ge 2d2 (observationally d = 3 d = 3 d=3d=3d=3); all proofs below are dimension-uniform, with d d ddd only in packing constants.
Information-measure class IM ( τ ) IM ( τ ) IM(tau)\mathsf{IM}(\tau)IM(τ). A divergence D D DDD lies in IM ( τ ) IM ( τ ) IM(tau)\mathsf{IM}(\tau)IM(τ) if: (i) data processing holds for GKSL channels and the pinching map P W P W P^(W)\mathcal{P}^WPW; (ii) LR quasi-factorization holds with constant C q ( τ ) C q ( τ ) C_(q)(tau)C_q(\tau)Cq(τ); (iii) a Dobrushin coefficient satisfies δ D ( P W Φ t ) e γ W t δ D ( P W Φ t ) e γ W t delta _(D)(P^(W)@Phi _(t)) <= e^(-gamma _(W)t)\delta_D(\mathcal{P}^W \circ \Phi_t) \le e^{-\gamma_W t}δD(PWΦt)eγWt at the KKT point; (iv) on calibrated sublevels there exist constants m ( τ ) , M ( τ ) m ( τ ) , M ( τ ) m(tau),M(tau)m(\tau), M(\tau)m(τ),M(τ) with m D 2 D M D 2 m D 2 D M D 2 mD_(2) <= D <= MD_(2)m D_2 \le D \le M D_2mD2DMD2 and similarly vs. KL. (Sandwiched Rényi 1 α 2 1 α 2 1 <= alpha <= 21 \le \alpha \le 21α2 and KL are in IM ( τ ) IM ( τ ) IM(tau)\mathsf{IM}(\tau)IM(τ).)
(H10.1) Fast dynamics & LR. On each finite union A A AAA, the fast sector evolves by a GKSL semigroup Φ t A = e t L A Φ t A = e t L A Phi_(t)^(A)=e^(tL^(A))\Phi_t^A = e^{t \mathcal{L}^A}ΦtA=etLA obeying Lieb–Robinson locality with velocity v L R v L R v_(LR)v_{\rm LR}vLR and profile f L R f L R f_(LR)f_{\rm LR}fLR, and respecting the three budget quadratics (Ch. 3).
(H10.2) Pointer alignment & Dirichlet linearity. The leakage quadratic is left-Dirichlet in the pointer weight W W WWW and orthogonally additive in its spectral decomposition; P W P W P^(W)\mathcal{P}^WPW is a D D DDD-contraction for any D IM D IM D inIMD \in \mathsf{IM}DIM.
(H10.3) Throughput normalization. = λ t h 1 = λ t h 1 ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1}=λth1 (Ch. 3), so the HS metric prices motion along unitary orbits.
(H10.4) Slow Γ-limit. Under localized Mosco/Attouch, first variations commute with ε 0 ε 0 epsi rarr0\varepsilon \to 0ε0 and the slow sector satisfies G μ ν + Λ g μ ν = 8 π G T μ ν e f f G μ ν + Λ g μ ν = 8 π G T μ ν e f f G_(mu nu)+Lambdag_(mu nu)=8pi GT_(mu nu)^(eff)G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G \, T^{\rm eff}_{\mu\nu}Gμν+Λgμν=8πGTμνeff, μ T μ ν e f f = 0 μ T μ ν e f f = 0 grad ^(mu)T_(mu nu)^(eff)=0\nabla^\mu T^{\rm eff}_{\mu\nu} = 0μTμνeff=0, with calibrated multipliers ( G , Λ ) ( G , Λ ) (G,Lambda)(G, \Lambda)(G,Λ).
Light-cone time. τ L R ( ) = / v L R τ L R ( ) = / v L R tau_(LR)(ℓ)=ℓ//v_(LR)\tau_{\rm LR}(\ell) = \ell / v_{\rm LR}τLR()=/vLR. The adjacency graph G G G_(ℓ)\mathcal{G}_\ellG connects face-sharing tiles.

10.0.1 Seed ensemble (poke process)

Pokes arrive as a Poisson random measure on spacetime with rate λ 0 λ 0 lambda_(0)\lambda_0λ0, amplitudes with sub-exponential tails (parameter θ > 0 θ > 0 theta > 0\theta > 0θ>0), and LR-compatible support (no super-cone events).
A seed tile is aligned if its post-poke state is P W P W P^(W)\mathcal{P}^WPW-close (in any D IM D IM D inIMD \in \mathsf{IM}DIM) to a pointer block. For γ W > 0 γ W > 0 gamma _(W) > 0\gamma_W > 0γW>0 and finite α l e a k α l e a k alpha_(leak)\alpha_{\rm leak}αleak, aligned seeds occur with positive probability.

10.1 Tile balance and the reproduction number

For a tile union A A AAA, let C t ( A ) C t ( A ) C_(t)(A)\mathsf{C}_t(A)Ct(A) be the coherence content at time t t ttt measured by any D IM D IM D inIMD \in \mathsf{IM}DIM. Over Δ t τ L R ( ) Δ t τ L R ( ) Delta t <= tau_(LR)(ℓ)\Delta t \le \tau_{\rm LR}(\ell)ΔtτLR(), three effects act on a seed tile T T TTT:
  1. Leakage load: L ( , Δ t ) C L α l e a k Δ t L ( , Δ t ) C L α l e a k Δ t L(ℓ,Delta t) <= C_(L)alpha_(leak)(Delta t)/(ℓ)L(\ell, \Delta t) \le C_L \, \alpha_{\rm leak} \, \tfrac{\Delta t}{\ell}L(,Δt)CLαleakΔt.
  2. Pointer alignment: η a l i g n ( Δ t ) = 1 δ D ( P W Φ Δ t ) ( 0 , 1 ] η a l i g n ( Δ t ) = 1 δ D ( P W Φ Δ t ) ( 0 , 1 ] eta_(align)(Delta t)=1-delta _(D)(P^(W)@Phi_(Delta t))in(0,1]\eta_{\rm align}(\Delta t) = 1 - \delta_D(\mathcal{P}^W \circ \Phi_{\Delta t}) \in (0, 1]ηalign(Δt)=1δD(PWΦΔt)(0,1] with 1 η a l i g n e γ W Δ t 1 η a l i g n e γ W Δ t 1-eta_(align) <= e^(-gamma _(W)Delta t)1 - \eta_{\rm align} \le e^{-\gamma_W \Delta t}1ηaligneγWΔt.
  3. Neighbor recruitment: M ( , Δ t ) C M ( v L R Δ t ) d M ( , Δ t ) C M ( v L R Δ t ) d M(ℓ,Delta t) <= C_(M)((v_(LR)Delta t)/(ℓ))^(d)M(\ell, \Delta t) \le C_M \, (\tfrac{v_{\rm LR} \Delta t}{\ell})^dM(,Δt)CM(vLRΔt)d (packing; App. I.1).
Definition 10.1 (Reproduction number). R c o h ( , Δ t ) = η a l i g n ( Δ t ) ( 1 L ( , Δ t ) ) M ( , Δ t ) R c o h ( , Δ t ) = η a l i g n ( Δ t ) ( 1 L ( , Δ t ) ) M ( , Δ t ) R_(coh)(ℓ,Delta t)=eta_(align)(Delta t)(1-L(ℓ,Delta t))M(ℓ,Delta t)\mathcal{R}_{\rm coh}(\ell, \Delta t) = \eta_{\rm align}(\Delta t) \, (1 - L(\ell, \Delta t)) \, M(\ell, \Delta t)Rcoh(,Δt)=ηalign(Δt)(1L(,Δt))M(,Δt).
Proposition 10.2 (Optimized one-step threshold). At Δ t = / v L R Δ t = / v L R Delta t=ℓ//v_(LR)\Delta t = \ell / v_{\rm LR}Δt=/vLR, R c o h ( 1 e γ W / v L R ) η ( 1 C L α l e a k / v L R ) 1 L C M M R c o h ( 1 e γ W / v L R ) η ( 1 C L α l e a k / v L R ) 1 L C M M R_(coh) >= ubrace((1-e^(-gamma _(W)ℓ//v_(LR)))ubrace)_(eta)ubrace((1-C_(L)alpha_(leak)//v_(LR))ubrace)_(1-L)ubrace(C_(M)ubrace)_(M)\mathcal{R}_{\rm coh} \ \ge \ \underbrace{(1 - e^{-\gamma_W \ell / v_{\rm LR}})}_{\eta} \, \underbrace{(1 - C_L \alpha_{\rm leak} / v_{\rm LR})}_{1 - L} \, \underbrace{C_M}_{M}Rcoh  (1eγW/vLR)η(1CLαleak/vLR)1LCMM.
Thus R c o h > 1 R c o h > 1 R_(coh) > 1\mathcal{R}_{\rm coh} > 1Rcoh>1 if ( 1 e γ W / v L R ) C M > ( 1 C L α l e a k / v L R ) 1 ( 1 e γ W / v L R ) C M > ( 1 C L α l e a k / v L R ) 1 (1-e^(-gamma _(W)ℓ//v_(LR)))C_(M) > (1-C_(L)alpha_(leak)//v_(LR))^(-1)(1 - e^{-\gamma_W \ell / v_{\rm LR}}) C_M > (1 - C_L \alpha_{\rm leak} / v_{\rm LR})^{-1}(1eγW/vLR)CM>(1CLαleak/vLR)1.

10.2 Nucleation and non-recurrence

Theorem 10.3 (Nucleation ⇒ supercritical cascade). If R c o h ( , Δ t ) > 1 R c o h ( , Δ t ) > 1 R_(coh)(ℓ,Delta t) > 1\mathcal{R}_{\rm coh}(\ell, \Delta t) > 1Rcoh(,Δt)>1 for some ( , Δ t ) ( , Δ t ) (ℓ,Delta t)(\ell, \Delta t)(,Δt), an aligned seed produces (in the worst-case envelope) a growing cluster of aligned tiles whose boundary advances at positive asymptotic speed. The occupied set stochastically dominates a supercritical Galton–Watson / first-passage process on G G G_(ℓ)\mathcal{G}_\ellG with mean offspring > 1 > 1 > 1>1>1.
Proof idea: LR quasi-factorization and orthogonal additivity in W W WWW yield FKG/Harris positive association; dominate by a branching process of mean R c o h > 1 R c o h > 1 R_(coh) > 1\mathcal{R}_{\rm coh} > 1Rcoh>1 (App. I.3).
Theorem 10.4 (Non-recurrence in aligned media). In an already pointer-aligned bath, R c o h a m b i e n t 1 R c o h a m b i e n t 1 R_(coh)^(ambient) <= 1\mathcal{R}_{\rm coh}^{\rm ambient} \le 1Rcohambient1 for all Δ t τ L R Δ t τ L R Delta t <= tau_(LR)\Delta t \le \tau_{\rm LR}ΔtτLR; seeds decohere or are absorbed.
Reason: DPI for P W e n v Φ Δ t P W e n v Φ Δ t P^(W_(env))@Phi_(Delta t)\mathcal{P}^{W_{\rm env}} \circ \Phi_{\Delta t}PWenvΦΔt gives η a l i g n e n v 1 η a l i g n e n v 1 eta_(align)^(env) <= 1\eta_{\rm align}^{\rm env} \le 1ηalignenv1 with equality only on the ambient block; leakage is at least that to the full bath.

10.2′ RG-stability in the tiling scale \ell

Rescale λ λ ℓ|->lambdaℓ\ell \mapsto \lambda \ellλ with λ ( 1 2 , 2 ) λ ( 1 2 , 2 ) lambda in((1)/(2),2)\lambda \in (\tfrac{1}{2}, 2)λ(12,2). Define the budget reparametrization α l e a k α l e a k ( λ ) := λ 1 α l e a k α l e a k α l e a k ( λ ) := λ 1 α l e a k alpha_(leak)|->alpha_(leak)^((lambda)):=lambda^(-1)alpha_(leak)\alpha_{\rm leak} \mapsto \alpha_{\rm leak}^{(\lambda)} := \lambda^{-1} \alpha_{\rm leak}αleakαleak(λ):=λ1αleak, α t h α t h ( λ ) := α t h α t h α t h ( λ ) := α t h alpha_(th)|->alpha_(th)^((lambda)):=alpha_(th)\alpha_{\rm th} \mapsto \alpha_{\rm th}^{(\lambda)} := \alpha_{\rm th}αthαth(λ):=αth, α c x α c x ( λ ) := α c x α c x α c x ( λ ) := α c x alpha_(cx)|->alpha_(cx)^((lambda)):=alpha_(cx)\alpha_{\rm cx} \mapsto \alpha_{\rm cx}^{(\lambda)} := \alpha_{\rm cx}αcxαcx(λ):=αcx, so that ρ l e a k α l e a k 1 ρ l e a k α l e a k 1 rho_(leak)propalpha_(leak)ℓ^(-1)\rho_{\rm leak} \propto \alpha_{\rm leak} \ell^{-1}ρleakαleak1 is invariant and the other budgets are unchanged to leading order.
Then to first order in | λ 1 | | λ 1 | |lambda-1||\lambda - 1||λ1|, observable quantities H ( a ) , w ( a ) , Ω k H ( a ) , w ( a ) , Ω k H(a),w(a),Omega _(k)H(a), w(a), \Omega_kH(a),w(a),Ωk are invariant up to controlled O ( | λ 1 | ) O ( | λ 1 | ) O(|lambda-1|)\mathcal{O}(|\lambda - 1|)O(|λ1|) shifts absorbed in calibration constants. (App. J.4 gives a beta-function view.)
Operational Planck tile. If v L R c v L R c v_(LR) <= cv_{\rm LR} \le cvLRc, adopt ~ P := P ( c / v L R ) 1 / 2 ~ P := P ( c / v L R ) 1 / 2 tilde(ℓ)_(P):=ℓ_(P)(c//v_(LR))^(1//2)\tilde{\ell}_P := \ell_P (c / v_{\rm LR})^{1/2}~P:=P(c/vLR)1/2 as the Planck-window coarse-graining scale; all Planck-window statements hold with ~ P ~ P ℓ~~ tilde(ℓ)_(P)\ell \approx \tilde{\ell}_P~P.

10.3 Budget quadratics ⇒ equations of state (with robustness)

Let the calibrated quadratics be: throughput Q t h ( A ) = [ H , A ] , [ H , A ] H S Q t h ( A ) = [ H , A ] , [ H , A ] H S Q_(th)(A)=(:[H,A],[H,A]:)_(HS)Q_{\rm th}(A) = \langle [H, A], [H, A] \rangle_{\rm HS}Qth(A)=[H,A],[H,A]HS; complexity Q c x ( A ) = ξ A , ξ A Q c x ( A ) = ξ A , ξ A Q_(cx)(A)=(:grad _(xi)A,grad _(xi)A:)Q_{\rm cx}(A) = \langle \nabla_\xi A, \nabla_\xi A \rangleQcx(A)=ξA,ξA (Ad-invariant Hilbertian with correlation length ξ ξ xi\xiξ); leakage Q l e a k ( A ) = ( I P W ) A 2 Q l e a k ( A ) = ( I P W ) A 2 Q_(leak)(A)=||(I-P^(W))A||^(2)Q_{\rm leak}(A) = \| (I - \mathcal{P}^W) A \|^2Qleak(A)=(IPW)A2 (Dirichlet in W W WWW). Denote multipliers α t h , α c x , α l e a k α t h , α c x , α l e a k alpha_(th),alpha_(cx),alpha_(leak)\alpha_{\rm th}, \alpha_{\rm cx}, \alpha_{\rm leak}αth,αcx,αleak.
Lemma 10.5 (Throughput). With = λ t h 1 = λ t h 1 ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1}=λth1: ρ t h = c t h α t h Ω 2 ρ t h = c t h α t h Ω 2 rho_(th)=c_(th)alpha_(th)Omega^(2)\rho_{\rm th} = c_{\rm th} \, \alpha_{\rm th} \, \Omega^2ρth=cthαthΩ2 and p t h = 1 3 ρ t h p t h = 1 3 ρ t h p_(th)=(1)/(3)rho_(th)p_{\rm th} = \tfrac{1}{3} \rho_{\rm th}pth=13ρth (ultra-relativistic sector).
Lemma 10.6 (Complexity). With tile-uniform LSI/Poincaré at scale ξ ξ xi\xiξ: ρ c x = c c x α c x ξ 2 ρ c x = c c x α c x ξ 2 rho_(cx)=c_(cx)alpha_(cx)xi^(-2)\rho_{\rm cx} = c_{\rm cx} \, \alpha_{\rm cx} \, \xi^{-2}ρcx=ccxαcxξ2, and p c x = ρ c x / ( 3 log a ) p c x = ρ c x / ( 3 log a ) p_(cx)=-delrho_(cx)//del(3log a)p_{\rm cx} = - \partial \rho_{\rm cx} / \partial (3 \log a)pcx=ρcx/(3loga).
Lemma 10.7 (Leakage). With left-Dirichlet Q l e a k Q l e a k Q_(leak)Q_{\rm leak}Qleak and pinching gap γ W γ W gamma _(W)\gamma_WγW: ρ l e a k = c l e a k α l e a k A / V c l e a k α l e a k 1 ρ l e a k = c l e a k α l e a k A / V c l e a k α l e a k 1 rho_(leak)=c_(leak)alpha_(leak)A//V∼c_(leak)alpha_(leak)ℓ^(-1)\rho_{\rm leak} = c_{\rm leak} \, \alpha_{\rm leak} \, A / V \sim c_{\rm leak} \, \alpha_{\rm leak} \, \ell^{-1}ρleak=cleakαleakA/Vcleakαleak1, and w l e a k [ 1 , 1 / 3 ] w l e a k [ 1 , 1 / 3 ] w_(leak)in[-1,1//3]w_{\rm leak} \in [-1, 1/3]wleak[1,1/3]. Endpoint w 1 w 1 w rarr-1w \to -1w1 holds when the leakage block is pointer-frozen (log-Sobolev gap γ 0 > 0 γ 0 > 0 >= gamma_(0) > 0\ge \gamma_0 > 0γ0>0); w 1 / 3 w 1 / 3 w rarr1//3w \to 1/3w1/3 holds if it thermalizes within the LR cone.
Stability under representatives. If Q Q QQQ and Q Q Q^(')Q'Q are norm-equivalent on the admissible class (constants m , M m , M m,Mm, Mm,M), the scalings and w w www persist; only c c c_(∙)c_\bulletc rescale within [ m , M ] [ m , M ] [m,M][m, M][m,M].

10.4 FRW emergence and quantitative near-flatness

Inside the supercritical domain Ω n u c Ω n u c Omega_(nuc)\Omega_{\rm nuc}Ωnuc, LR locality and D D DDD-contraction imply decay of anisotropic stress and vorticity over large boxes B R B R B_(R)B_RBR: π i j + ω i C | B R | | B R | Φ ( α l e a k α t h , α l e a k α c x ) R 0 π i j + ω i C | B R | | B R | Φ α l e a k α t h , α l e a k α c x R 0 ||pi_(ij)||+||omega _(i)|| <= C(|delB_(R)|)/(|B_(R)|)Phi((alpha_(leak))/(alpha_(th)),(alpha_(leak))/(alpha_(cx)))rarr"R rarr oo"0\| \pi_{ij} \| + \| \omega_i \| \ \le \ C \, \frac{|\partial B_R|}{|B_R|} \, \Phi \! \left( \tfrac{\alpha_{\rm leak}}{\alpha_{\rm th}}, \tfrac{\alpha_{\rm leak}}{\alpha_{\rm cx}} \right) \xrightarrow{R \to \infty} 0πij+ωi  C|BR||BR|Φ(αleakαth,αleakαcx)R0.
Thus the slow Γ-limit reduces to FRW with p e f f = 1 3 ρ e f f + o ( 1 ) p e f f = 1 3 ρ e f f + o ( 1 ) p_(eff)=(1)/(3)rho_(eff)+o(1)p_{\rm eff} = \tfrac{1}{3} \rho_{\rm eff} + o(1)peff=13ρeff+o(1) initially and standard Friedmann–Raychaudhuri holds.
Proposition 10.8 (Near-flatness inequality). For any comoving exhaustion B R B R B_(R)B_RBR and horizon time t H ( R ) t H ( R ) t_(H)(R)t_H(R)tH(R), sup t t H ( R ) | K | a 2 C f l a t α l e a k A ( B R ) V ( B R ) H 2 + o R ( 1 ) sup t t H ( R ) | K | a 2 C f l a t α l e a k A ( B R ) V ( B R ) H 2 + o R ( 1 ) s u p_(t <= t_(H)(R))(|K|)/(a^(2)) <= C_(flat)(alpha_(leak)A(B_(R)))/(V(B_(R))H^(2))+o_(R)(1)\sup_{t \le t_H(R)} \frac{|K|}{a^2} \ \le \ C_{\rm flat} \, \frac{\alpha_{\rm leak} \, A(B_R)}{V(B_R) \, H^2} + o_R(1)supttH(R)|K|a2  CflatαleakA(BR)V(BR)H2+oR(1). Equivalently, | Ω k | C f l a t ( α l e a k / α t h ) ( A / V ) ( Ω r / H 0 2 ) + o R ( 1 ) | Ω k | C f l a t ( α l e a k / α t h ) ( A / V ) ( Ω r / H 0 2 ) + o R ( 1 ) |Omega _(k)| <= C_(flat)(alpha_(leak)//alpha_(th))(A//V)(Omega _(r)//H_(0)^(2))+o_(R)(1)|\Omega_k| \le C_{\rm flat} (\alpha_{\rm leak} / \alpha_{\rm th}) (A / V) (\Omega_r / H_0^2) + o_R(1)|Ωk|Cflat(αleak/αth)(A/V)(Ωr/H02)+oR(1).

10.5 Risk-sensitive envelope; epi-convergence and commuting limits

Let ( P , F , μ β ) ( P , F , μ β ) (P,F,mu _(beta))(\mathcal{P}, \mathcal{F}, \mu_\beta)(P,F,μβ) be the poke space with exponentially tight tails. Define CL ( A , Φ ) = inf p P L ( A , Φ ; p ) CL ( A , Φ ) = inf p P L ( A , Φ ; p ) CL(A,Phi)=i n f_(p inP)L(A,Phi;p)\mathrm{CL}(A, \Phi) = \inf_{p \in \mathcal{P}} L(A, \Phi; p)CL(A,Φ)=infpPL(A,Φ;p), l.s.c. in ( A , Φ ) ( A , Φ ) (A,Phi)(A, \Phi)(A,Φ).
Lemma 10.9 (Envelope epi-convergence). As β β beta rarr oo\beta \to \inftyβ with μ β μ μ β μ mu _(beta)=>mu _(oo)\mu_\beta \Rightarrow \mu_\inftyμβμ supported on the admissible cone, lim β sup A CL β ( A , Φ ) = sup A ess inf p μ L ( A , Φ ; p ) lim β sup A CL β ( A , Φ ) = sup A ess inf p μ L ( A , Φ ; p ) lim_(beta rarr oo)s u p _(A)CL_(beta)(A,Phi)=s u p _(A)essinf_(p∼mu _(oo))L(A,Phi;p)\lim_{\beta \to \infty} \sup_A \mathrm{CL}_\beta(A, \Phi) = \sup_A \operatorname*{ess\,inf}_{p \sim \mu_\infty} L(A, \Phi; p)limβsupACLβ(A,Φ)=supAessinfpμL(A,Φ;p), and maximizers exist on calibrated sublevels.
Lemma 10.10 (Commuting limits). On finite boxes, taking β β beta rarr oo\beta \to \inftyβ then ε 0 ε 0 epsi rarr0\varepsilon \to 0ε0 (Γ-limit) yields the same envelope as any interleaving respecting LR locality.

10.6 Conservation in the slow sector (Noether + Γ-limit)

Let the slow functional be S [ g ; α ] S [ g ; α ] S[g;alpha_(∙)]\mathcal{S}[g; \alpha_\bullet]S[g;α] with budget constraints enforced by multipliers. Variations δ g δ g delta g\delta gδg obeying compact support and cone-preserving gauge produce δ S = 1 2 ( T μ ν e f f ) δ g μ ν g d 4 x δ S = 1 2 ( T μ ν e f f ) δ g μ ν g d 4 x deltaS=(1)/(2)int(T_(mu nu)^(eff))deltag^(mu nu)sqrt(-g)d^(4)x\delta \mathcal{S} = \tfrac{1}{2} \int (T^{\rm eff}_{\mu\nu}) \, \delta g^{\mu\nu} \, \sqrt{-g} \, d^4 xδS=12(Tμνeff)δgμνgd4x, T e f f = T v i s + T f a s t T e f f = T v i s + T f a s t T^(eff)=T^(vis)+T^(fast)T^{\rm eff} = T^{\rm vis} + T^{\rm fast}Teff=Tvis+Tfast. Cone-preserving diffeomorphisms give T e f f = 0 T e f f = 0 grad*T^(eff)=0\nabla \cdot T^{\rm eff} = 0Teff=0. Leakage only exchanges budget within T f a s t T f a s t T^(fast)T^{\rm fast}Tfast; the sum is conserved in the slow sector.

10.7 Big-Bang vs. coherence bounce; energy conditions

Let ρ = ρ t h + ρ c x + ρ l e a k ρ = ρ t h + ρ c x + ρ l e a k rho=rho_(th)+rho_(cx)+rho_(leak)\rho = \rho_{\rm th} + \rho_{\rm cx} + \rho_{\rm leak}ρ=ρth+ρcx+ρleak with w t h 1 3 w t h 1 3 w_(th)~~(1)/(3)w_{\rm th} \approx \tfrac{1}{3}wth13, w c x [ 0 , 1 ] w c x [ 0 , 1 ] w_(cx)in[0,1]w_{\rm cx} \in [0, 1]wcx[0,1], w l e a k [ 1 , 1 / 3 ] w l e a k [ 1 , 1 / 3 ] w_(leak)in[-1,1//3]w_{\rm leak} \in [-1, 1/3]wleak[1,1/3].
Theorem 10.11 (Bounce criterion). If ρ l e a k + 3 p l e a k < ( ρ t h + ρ c x + 3 p t h + 3 p c x ) ρ l e a k + 3 p l e a k < ( ρ t h + ρ c x + 3 p t h + 3 p c x ) rho_(leak)+3p_(leak) < -(rho_(th)+rho_(cx)+3p_(th)+3p_(cx))\rho_{\rm leak} + 3 p_{\rm leak} < - (\rho_{\rm th} + \rho_{\rm cx} + 3 p_{\rm th} + 3 p_{\rm cx})ρleak+3pleak<(ρth+ρcx+3pth+3pcx) over an interval, then a ¨ > 0 a ¨ > 0 a^(¨) > 0\ddot{a} > 0a¨>0 and a minimum scale factor a min > 0 a min > 0 a_(min) > 0a_{\min} > 0amin>0 occurs. (Raychaudhuri.)
Lemma 10.12 (SEC/ANEC status). The effective fluid need not satisfy SEC; pointer-frozen leakage ( w 1 w 1 w rarr-1w \to -1w1) violates SEC while LR microcausality holds. Hence singularity theorems’ hypotheses fail in the bounce branch.
Sequestered windows; anisotropy seeds. Near horizons/interiors, leakage throttling ( L L L darrL \downarrowL) with γ W γ W gamma _(W)uarr\gamma_W \uparrowγW can allow R c o h > 1 R c o h > 1 R_(coh) > 1\mathcal{R}_{\rm coh} > 1Rcoh>1 inside a causal window, producing daughter domains that are sequestered (domain-wall containment). Front roughness induces a two-point function δ ρ ( x ) δ ρ ( y ) r ( d 1 ) δ ρ ( x ) δ ρ ( y ) r ( d 1 ) (:delta rho(x)delta rho(y):)∼r^(-(d-1))\langle \delta \rho(\mathbf{x}) \delta \rho(\mathbf{y}) \rangle \sim r^{-(d-1)}δρ(x)δρ(y)r(d1) inside the LR horizon, giving a near scale-invariant spectrum in d = 3 d = 3 d=3d=3d=3 after horizon crossing (cf. Sim-S2).

10.8 Calibration protocol; constants; falsifiers

Constants C L , γ W , v L R , C M , c C L , γ W , v L R , C M , c C_(L),gamma _(W),v_(LR),C_(M),c_(∙)C_L, \gamma_W, v_{\rm LR}, C_M, c_\bulletCL,γW,vLR,CM,c are fixed by App. J.2; predictions are stable under norm-equivalent representatives (App. J.1).
Falsifiers: (i) LR violation; (ii) failure of Lemma 10.9 (non-attainment); (iii) robust violation of Prop. 10.8 across multiple B R B R B_(R)B_RBR at fixed calibration; (iv) in-medium supercritical cascades (violates Thm 10.4); (v) need for a fourth budget.

Chapter 11 — Planck Window: Budgets, Area Bounds, and Operational Scales

Scope. We fix Planck-window hypotheses, prove short-time and global-in-time tile-area bounds for cross-boundary information in any D IM ( τ ) D IM ( τ ) D inIM(tau)D \in \mathsf{IM}(\tau)DIM(τ), and relate the tiling scale to the operational Planck length ~ P ~ P tilde(ℓ)_(P)\tilde{\ell}_P~P when v L R c v L R c v_(LR) <= cv_{\rm LR} \le cvLRc.

11.1 Planck-window hypotheses

  • (P1) Local dimension control. On any tile T T TTT of side \ell, d l o c ( ) ( / ) κ d l o c ( ) ( / ) κ d_(loc)(ℓ)≲(ℓ//ℓ_(**))^(kappa)d_{\rm loc}(\ell) \lesssim (\ell / \ell_*)^{\kappa}dloc()(/)κ for microscopic ℓ_(**)\ell_*, κ > 0 κ > 0 kappa > 0\kappa > 0κ>0.
  • (P2) LR microcausality. GKSL obeys LR bounds with velocity v L R ( ) v L R ( ) v_(LR)(ℓ)v_{\rm LR}(\ell)vLR() uniformly finite across tiles.
  • (P3) Dirichlet linearity. Leakage is left-Dirichlet in W W WWW and orthogonally additive in its spectral decomposition.
  • (P4) Throughput normalization. = λ t h 1 = λ t h 1 ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1}=λth1 (local blocks and inductive limit).
  • (P5) Information class. Use D IM ( τ ) D IM ( τ ) D inIM(tau)D \in \mathsf{IM}(\tau)DIM(τ) throughout.
Operational Planck tile. If v L R c v L R c v_(LR) <= cv_{\rm LR} \le cvLRc, the Planck-window tile is ~ P = P ( c / v L R ) 1 / 2 ~ P = P ( c / v L R ) 1 / 2 tilde(ℓ)_(P)=ℓ_(P)(c//v_(LR))^(1//2)\tilde{\ell}_P = \ell_P (c / v_{\rm LR})^{1/2}~P=P(c/vLR)1/2; use ~ P ~ P ℓ~~ tilde(ℓ)_(P)\ell \approx \tilde{\ell}_P~P in area bounds.

11.2 Short-time tile-area bound (single cone)

For a union A A AAA of tiles and times t c t i m e / v L R t c t i m e / v L R t <= c_(time)ℓ//v_(LR)t \le c_{\rm time} \, \ell / v_{\rm LR}tctime/vLR, the GKSL evolution at a KKT point satisfies, for any D IM ( τ ) D IM ( τ ) D inIM(tau)D \in \mathsf{IM}(\tau)DIM(τ),
I D ( A : A ¯ ; t ) C i n f o ( D ) | A | Φ ( α l e a k α t h , α l e a k α c x ) , I D ( A : A ¯ ; t ) C i n f o ( D ) | A | Φ α l e a k α t h , α l e a k α c x , I_(D)(A: bar(A);t) <= C_(info)(D)(|del A|)/(ℓ)Phi((alpha_(leak))/(alpha_(th)),(alpha_(leak))/(alpha_(cx))),\boxed{ I_D(A : \bar{A}; t) \ \le \ C_{\rm info}(D) \, \frac{|\partial A|}{\ell} \, \Phi \! \left( \frac{\alpha_{\rm leak}}{\alpha_{\rm th}}, \frac{\alpha_{\rm leak}}{\alpha_{\rm cx}} \right) },ID(A:A¯;t)  Cinfo(D)|A|Φ(αleakαth,αleakαcx),
with C i n f o ( D ) = C 0 ( D ) ( 1 e γ W t ) / ( 1 e γ t ) C i n f o ( D ) = C 0 ( D ) ( 1 e γ W t ) / ( 1 e γ t ) C_(info)(D)=C_(0)(D)(1-e^(-gamma _(W)t))//(1-e^(-gamma_(**)t))C_{\rm info}(D) = C_0(D) \, (1 - e^{-\gamma_W t}) / (1 - e^{-\gamma_* t})Cinfo(D)=C0(D)(1eγWt)/(1eγt). Here γ γ gamma_(**)\gamma_*γ is the slowest mixing rate in A A AAA; c t i m e , C 0 ( D ) c t i m e , C 0 ( D ) c_(time),C_(0)(D)c_{\rm time}, C_0(D)ctime,C0(D) depend only on LR and tile LSI.
Proof sketch: LR quasi-factorization + pinching contraction + orthogonal additivity in W W WWW. Constants are calibrated, not universal.

11.2′ Globalized area bound (all times)

For any t 0 t 0 t >= 0t \ge 0t0,
I D ( A : A ¯ ; t ) | A | [ C 1 ( D ) ( 1 e γ W t ) + C 2 ( D ) 0 t e γ ( t s ) Ξ ( s ) d s ] , I D ( A : A ¯ ; t ) | A | [ C 1 ( D ) ( 1 e γ W t ) + C 2 ( D ) 0 t e γ ( t s ) Ξ ( s ) d s ] , I_(D)(A: bar(A);t) <= (|del A|)/(ℓ)[C_(1)(D)(1-e^(-gamma _(W)t))+C_(2)(D)int_(0)^(t)e^(-gamma_(**)(t-s))Xi(s)ds],I_D(A : \bar{A}; t) \ \le \ \frac{|\partial A|}{\ell} \Big[ C_1(D) \big( 1 - e^{-\gamma_W t} \big) + C_2(D) \! \int_0^t e^{-\gamma_* (t - s)} \, \Xi(s) \, ds \Big],ID(A:A¯;t)  |A|[C1(D)(1eγWt)+C2(D)0teγ(ts)Ξ(s)ds],
where Ξ Ξ Xi\XiΞ depends only on LR profile and tile LSI constants. (Iterated cone decomposition + Grönwall on I D I D I_(D)I_DID.)
Consequence. At ~ P ~ P ℓ~~ tilde(ℓ)_(P)\ell \approx \tilde{\ell}_P~P, cross-boundary predictive content is area-limited with coefficients controlled by budget ratios and ( γ W , v L R , LSI ) ( γ W , v L R , LSI ) (gamma _(W),v_(LR),"LSI")(\gamma_W, v_{\rm LR}, \text{LSI})(γW,vLR,LSI) fixed at calibration; stability holds under norm-equivalent representatives (App. J.1).

11.3 Inflationary embedding (optional)

If an inflaton exists, coherence supplies dissipative slow-roll corrections: leakage shifts friction by Δ Γ α l e a k Δ Γ α l e a k Delta Gamma propalpha_(leak)\Delta \Gamma \propto \alpha_{\rm leak}ΔΓαleak, throughput enforces a kinetic penalty α t h α t h propalpha_(th)\propto \alpha_{\rm th}αth; LR locality is preserved. Pure coherence-smoothing (§10.4) remains viable with T v i s T v i s T^(vis)T^{\rm vis}Tvis absent.

Chapter 12 — Dark Sector from Coherence (Minimal Candidates + Inequalities)

Scope. We present three minimal dark-matter candidates as coherence-selected patterns. Each couples gravitationally via the slow sector, is budget-suppressed in the fast/pointer sector, and admits quantitative inequalities mapping viability directly to calibrated α α alpha\alphaα-ratios and constants.

12.1 Hidden-pointer (HP) sector — CDM-like

Construction. Let W = a w a P a W = a w a P a W=sum _(a)w_(a)P_(a)W = \sum_a w_a P_aW=awaPa; a hidden block P D P D P_(D)P_DPD with w D w v i s w D w v i s w_(D)≫w_(vis)w_D \gg w_{\rm vis}wDwvis suppresses leakage for operators on P D P D P_(D)P_DPD. Consider a HP U(1) sector with field strength F D F D F^(D)F^DFD contributing only gravitationally at the Γ-limit.
Coldness inequality. The effective sound speed obeys c s 2 ε c o l d := C g a p ( T Δ H P ) 2 c s 2 ε c o l d := C g a p ( T Δ H P ) 2 c_(s)^(2) <= epsi_(cold):=C_(gap)((T)/(Delta_(HP)))^(2)c_s^2 \ \le \ \varepsilon_{\rm cold} := C_{\rm gap} \Big( \frac{T}{\Delta_{\rm HP}} \Big)^2cs2  εcold:=Cgap(TΔHP)2, with pointer gap Δ H P Δ H P Delta_(HP)\Delta_{\rm HP}ΔHP. Require c s 2 10 6 c s 2 10 6 c_(s)^(2)≲10^(-6)c_s^2 \lesssim 10^{-6}cs2106 by matter–radiation equality.
Discriminants. (i) Growth-index shift Δ γ Δ γ Delta gamma\Delta \gammaΔγ from residual c s 2 0 c s 2 0 c_(s)^(2)!=0c_s^2 \neq 0cs20; (ii) absence of isocurvature by pointer isolation. Failure of either falsifies HP under fixed calibration.

12.2 Phase-mode axion (PX) — misalignment cold/warm

Construction. A global U(1) phase, protected by pointer alignment, yields a light phase mode θ θ theta\thetaθ with V ( θ ) = Λ c x 4 ( 1 cos θ ) V ( θ ) = Λ c x 4 ( 1 cos θ ) V(theta)=Lambda_(cx)^(4)(1-cos theta)V(\theta) = \Lambda_{\rm cx}^4 \, (1 - \cos \theta)V(θ)=Λcx4(1cosθ), Λ c x 4 α c x Λ c x 4 α c x Lambda_(cx)^(4)propalpha_(cx)\Lambda_{\rm cx}^4 \propto \alpha_{\rm cx}Λcx4αcx, kinetic term fixed by = λ t h 1 = λ t h 1 ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1}=λth1. Misalignment abundance ρ P X ( a 0 ) = 1 2 f a 2 m a 2 θ 0 2 T ( m a / H ) ρ P X ( a 0 ) = 1 2 f a 2 m a 2 θ 0 2 T ( m a / H ) rho_(PX)(a_(0))=(1)/(2)f_(a)^(2)m_(a)^(2)theta_(0)^(2)T(m_(a)//H)\rho_{\rm PX}(a_0) = \tfrac{1}{2} f_a^2 m_a^2 \, \theta_0^2 \, \mathcal{T} \big( m_a / H \big)ρPX(a0)=12fa2ma2θ02T(ma/H), with ( m a , f a ) ( m a , f a ) (m_(a),f_(a))(m_a, f_a)(ma,fa) constrained by ( α c x , α t h ) ( α c x , α t h ) (alpha_(cx),alpha_(th))(\alpha_{\rm cx}, \alpha_{\rm th})(αcx,αth) calibration.
Warmness bound. Ly- α α alpha\alphaα and LSS require free-streaming below the observed cutoff, which translates to a lower bound on m a m a m_(a)m_ama (given f a f a f_(a)f_afa) and excludes parts of ( α c x , α t h ) ( α c x , α t h ) (alpha_(cx),alpha_(th))(\alpha_{\rm cx}, \alpha_{\rm th})(αcx,αth) space.
Discriminant. A correlated late-time viscosity from Q c x Q c x Q_(cx)Q_{\rm cx}Qcx must align with the allowed ( m a , f a ) ( m a , f a ) (m_(a),f_(a))(m_a, f_a)(ma,fa); mismatch falsifies PX.

12.3 Sterile-mixing neutrino (SN) — warm

Construction. A minimal seesaw-like block with derivation-pricing + leakage alignment suppresses visible mixing by sin 2 ( 2 θ ) C m i x ( α t h α l e a k ) 2 sin 2 ( 2 θ ) C m i x ( α t h α l e a k ) 2 sin^(2)(2theta)∼C_(mix)((alpha_(th))/(alpha_(leak)))^(2)\sin^2(2\theta) \ \sim \ C_{\rm mix} \Big( \frac{\alpha_{\rm th}}{\alpha_{\rm leak}} \Big)^2sin2(2θ)  Cmix(αthαleak)2. Freeze-in from pointer-misaligned dissipators sets the relic abundance.
Free-streaming inequality. Require λ F S v ( a ) a 2 H ( a ) d a 0.1 Mpc λ F S v ( a ) a 2 H ( a ) d a 0.1 Mpc lambda_(FS)≃int(v(a))/(a^(2)H(a))da <= 0.1"Mpc"\lambda_{\rm FS} \ \simeq \ \int \frac{v(a)}{a^2 H(a)} \, da \ \le \ 0.1 \ \text{Mpc}λFS  v(a)a2H(a)da  0.1 Mpc, which imposes a lower bound on mixing mapped to α t h / α l e a k α t h / α l e a k alpha_(th)//alpha_(leak)\alpha_{\rm th} / \alpha_{\rm leak}αth/αleak; combine with X-ray line limits to confine the viable band.

12.4 Shared falsifiers; calibration stability; baryogenesis routes

All candidates share a single calibration ( α t h , α c x , α l e a k ) ( α t h , α c x , α l e a k ) (alpha_(th),alpha_(cx),alpha_(leak))(\alpha_{\rm th}, \alpha_{\rm cx}, \alpha_{\rm leak})(αth,αcx,αleak). A persistent need for a fourth budget, or lab evidence of basis-invariant decoherence contradicting pointer alignment, invalidates the constructions.
Proposition 12.1 (Calibration stability). If , , (:*,*:)\langle \cdot, \cdot \rangle, and , , (:*,*:)^(')\langle \cdot, \cdot \rangle', are norm-equivalent on the admissible class with m X 2 X 2 M X 2 m X 2 X 2 M X 2 m||X||^(2) <= ||X||^('2) <= M||X||^(2)m \| X \|^2 \le \| X \|'^2 \le M \| X \|^2mX2X2MX2, then multipliers satisfy m α α M α m α α M α malpha_(∙) <= alpha_(∙)^(') <= Malpha_(∙)m \, \alpha_\bullet \le \alpha'_\bullet \le M \, \alpha_\bulletmααMα component-wise; predictions depending on ratios α i / α j α i / α j alpha _(i)//alpha _(j)\alpha_i / \alpha_jαi/αj are invariant up to [ m , M ] [ m , M ] [m,M][m, M][m,M].
Baryogenesis. Baseline: standard leptogenesis via the SN sector. Optional: leakage-phase baryogenesis, where a pointer-dependent phase in the Dirichlet form biases sphalerons (predicts CP-odd leakage observables in lab GKSL analogs).

Chapter 13 — Predictions, Calibration & Tests

Scope. Testable predictions with one primary KPI each, minimal supports, and calibration procedures. Constants are linked to multipliers via envelope identities. Global KPI carried through. We fit the coherence number χ = τ d e c / τ m e s s χ = τ d e c / τ m e s s chi=tau_(dec)//tau_(mess)\chi = \tau_{\rm dec} / \tau_{\rm mess}χ=τdec/τmess in each setting and extract multipliers via the envelope identities (App. E.4). Each subsection names one predictive bit (primary KPI) and at most two supports; “ablation” notes indicate how relaxing assumptions would alter the fit.

13.0 Toy-world validation of CL CL CL\mathrm{CL}CL (CA and qubit)

Protocol. We run the CA toy ( CL C A CL C A CL_(CA)\mathrm{CL}_{\rm CA}CLCA) on n × n n × n n xx nn \times nn×n grids with a fixed menu T C A T C A T_(CA)\mathscr{T}_{\rm CA}TCA of thresholds and weights, and the qubit toy ( CL Q CL Q CL_(Q)\mathrm{CL}_{\rm Q}CLQ) with ( ρ 0 , ρ 1 ) ( ρ 0 , ρ 1 ) (rho_(0),rho_(1))(\rho_0, \rho_1)(ρ0,ρ1) aligned/misaligned with W W WWW. For each, we estimate the global KPI χ = τ d e c / τ m e s s χ = τ d e c / τ m e s s chi=tau_(dec)//tau_(mess)\chi = \tau_{\rm dec} / \tau_{\rm mess}χ=τdec/τmess and fit multipliers via the envelope identities.
Predicted invariants. (i) Monotone equivalence: rankings of A A AAA by CL C A CL C A CL_(CA)\mathrm{CL}_{\rm CA}CLCA and CL Q CL Q CL_(Q)\mathrm{CL}_{\rm Q}CLQ agree up to an increasing transform; (ii) Pointer alignment: leakage-minimizing generators align with W W WWW; (iii) Ablation: relaxing Ad-invariance reintroduces a fourth quadratic direction and breaks the equivalence (Section 2.2 note).
Measurement. Report CL CL CL\mathrm{CL}CL-scores with 95 % 95 % 95%95\%95% binomial CIs, estimated χ χ chi\chiχ, and inferred multipliers. Each run uses finite, auditable protocols with reproducible seeds.

13.1 Gravitational-wave phasing residual (binary inspirals)

Prediction. Phase residual Δ p Δ p Delta p\Delta pΔp relative to GR templates scales as Δ p ε η ( π G M c f / c 3 ) η Δ p ε η ( π G M c f / c 3 ) η Delta p~~epsieta(pi GM_(c)f//c^(3))^(eta)\Delta p \approx \varepsilon \, \eta \, (\pi G M_c f / c^3)^{\eta}Δpεη(πGMcf/c3)η with ε 0.05 , η 1 ε 0.05 , η 1 epsi∼0.05,eta~~1\varepsilon \sim 0.05, \eta \approx 1ε0.05,η1.
KPI: bias-robust estimator of η η eta\etaη across events.
Supports: residual-vs-frequency slope; cross-event scaling with M c M c M_(c)M_cMc.

13.2 Horizon flux suppression

Prediction. Outgoing flux below Hawking prediction by factor f ( λ l e a k , W ) 1 f ( λ l e a k , W ) 1 f(lambda_(leak),W) <= 1f(\lambda_{\rm leak}, W) \le 1f(λleak,W)1.
KPI: F o u t / F H a w k i n g F o u t / F H a w k i n g F_(out)//F_(Hawking)\mathcal{F}_{\rm out} / \mathcal{F}_{\rm Hawking}Fout/FHawking.
Supports: two-point distortions; pointer-basis stability.

13.3 Pointer basis in noisy interferometers

Prediction. Environmental weight W W WWW selects a unique pointer basis minimizing j ω ( L j W L j ) j ω ( L j W L j ) sum _(j)omega(L_(j)^(†)WL_(j))\sum_j \omega(L_j^\dagger W L_j)jω(LjWLj).
KPI: basis-aligned decoherence rates vs misaligned configs.

13.4 Calibration of \hbar

With A ˙ = i 1 [ H , A ] A ˙ = i 1 [ H , A ] A^(˙)=iℏ^(-1)[H,A]\dot{A} = i \hbar^{-1} [H, A]A˙=i1[H,A], use a two-level system at frequency Ω Ω Omega\OmegaΩ; measure Fubini–Study speed v v vvv; set v = 2 Δ H / v = 2 Δ H / v=2Delta H//ℏv = 2 \Delta H / \hbarv=2ΔH/ to calibrate = λ t h 1 = λ t h 1 ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1}=λth1.

13.5 Envelope identities

Optimal value Φ ( τ ) Φ ( τ ) Phi(tau)\Phi(\tau)Φ(τ) λ = Φ / τ λ = Φ / τ lambda_(∙)=del Phi//deltau_(∙)\lambda_\bullet = \partial \Phi / \partial \tau_\bulletλ=Φ/τ a.e.; defines , G , Λ , α i , , G , Λ , α i , ℏ,G,Lambda,alpha _(i),dots\hbar, G, \Lambda, \alpha_i, \ldots,G,Λ,αi,.
Protocol: fit λ λ lambda\lambdaλs by matching observed invariants under RG constraints.

13.6 Measurement hygiene

Collect only predictive signals; cone-limited timing windows; publish priors/instruments/code; pre-register stop rules.

Appendix A — H0′: Spaces, Norms & Budgets (C*-compatible)

  • Algebra. Inductive-limit C*-algebra A = Λ A Λ A = Λ A Λ ¯ A= bar(uuu _(Lambda)A_(Lambda))\mathcal{A} = \overline{\bigcup_\Lambda \mathcal{A}_\Lambda}A=ΛAΛ.
  • State/GNS. Faithful normal state ω ω omega\omegaω, GNS triple ( π ω , H ω , Ω ω ) ( π ω , H ω , Ω ω ) (pi _(omega),H_(omega),Omega _(omega))(\pi_\omega, \mathcal{H}_\omega, \Omega_\omega)(πω,Hω,Ωω); identify A A A\mathcal{A}A with π ω ( A ) B ( H ω ) π ω ( A ) B ( H ω ) pi _(omega)(A)subB(H_(omega))\pi_\omega(\mathcal{A}) \subset \mathcal{B}(\mathcal{H}_\omega)πω(A)B(Hω).
  • Core. Dense invariant D D D\mathcal{D}D common to all generators.
Throughput quadratic. On each finite Λ Λ Lambda\LambdaΛ, using normalized HS inner product from tr Λ / d Λ tr Λ / d Λ tr_(Lambda)//d_( Lambda)\mathrm{tr}_\Lambda / d_\LambdatrΛ/dΛ:
B t h Λ := 1 2 δ H d e r ; Λ 2 , B t h := sup Λ B t h Λ . B t h Λ := 1 2 δ H d e r ; Λ 2 , B t h := sup Λ B t h Λ . B_(th)^(Lambda):=(1)/(2)||delta _(H)||_(der;Lambda)^(2),quadB_(th):=s u p _(Lambda)B_(th)^(Lambda).\mathcal{B}_{\rm th}^\Lambda := \tfrac{1}{2} \| \delta_H \|_{{\rm der}; \Lambda}^2, \quad \boxed{ \mathcal{B}_{\rm th} := \sup_\Lambda \mathcal{B}_{\rm th}^\Lambda }.BthΛ:=12δHder;Λ2,Bth:=supΛBthΛ.
Leakage budget.
B l e a k ( { L j } , W ) = j W 1 / 2 L j 2 , ω 2 = j ω ( L j W L j ) B l e a k ( { L j } , W ) = j W 1 / 2 L j 2 , ω 2 = j ω ( L j W L j ) B_(leak)({L_(j)},W)=sum _(j)||W^(1//2)L_(j)||_(2,omega)^(2)=sum _(j)omega(L_(j)^(†)WL_(j))\boxed{ \, \mathcal{B}_{\rm leak}(\{L_j\}, W) = \sum_j \| W^{1/2} L_j \|_{2, \omega}^2 = \sum_j \omega(L_j^\dagger W L_j) \, }Bleak({Lj},W)=jW1/2Lj2,ω2=jω(LjWLj)
Complexity budget.
B c x ( L ) = inf { α κ α J α c b 2 : L = α J α on A l o c } B c x ( L ) = inf { α κ α J α c b 2 : L = α J α  on  A l o c } B_(cx)(L)=i n f{sum _(alpha)kappa _(alpha)||J_(alpha)||_(cb)^(2):L=sum _(alpha)J_(alpha)" on "A_(loc)}\boxed{ \, \mathcal{B}_{\rm cx}(\mathcal{L}) = \inf \Big\{ \sum_\alpha \kappa_\alpha \| \mathcal{J}_\alpha \|_{cb}^2 : \ \mathcal{L} = \sum_\alpha \mathcal{J}_\alpha \ \text{ on } \ \mathcal{A}_{\rm loc} \Big\} \, }Bcx(L)=inf{ακαJαcb2: L=αJα  on  Aloc}
Collected properties. B t h , B l e a k , B c x B t h , B l e a k , B c x B_(th),B_(leak),B_(cx)\mathcal{B}_{\rm th}, \mathcal{B}_{\rm leak}, \mathcal{B}_{\rm cx}Bth,Bleak,Bcx are convex, l.s.c.; sublevels equi-coercive (Ch. 4). Only three independent budgets up to nulls (Thm I.1).
Notation. We use B c x B c x B_(cx)B_{\rm cx}Bcx (“complexity”) interchangeably with the local-decomposition compression quadratic; B c x B c x B c x B c x B_(cx)-=B_(cx)B_{\rm cx} \equiv \mathcal{B}_{\rm cx}BcxBcx.

Appendix B — Large-Deviation Machinery (Risk-Sensitive ⇒ Worst-Case)

  • Varadhan’s lemma for log E e β log E e β log Ee^(betaℓ)\log \mathbb{E} e^{\beta \ell}logEeβ β β beta rarr oo\beta \to \inftyβ yields worst-case inner min.
  • Laplace principle on A A A\mathcal{A}A under coercivity and u.s.c.
  • Epi-convergence in β β beta\betaβ: CL β inf Φ CL CL β inf Φ CL CL_(beta)↘i n f _(Phi)CL\mathrm{CL}_\beta \searrow \inf_\Phi \mathrm{CL}CLβinfΦCL.

Appendix C — Gauge/Matter Derivations (Yukawas, Anomalies, Hypercharge)

Binder set, linear system, and solution giving the SM hypercharges (details as in Ch. 6).

Appendix D — Γ-Limit to Einstein–Hilbert (airtight revision)

Goal. Construct an explicit, gauge-fixed family { F ε } ε 0 { F ε } ε 0 {F_(epsi)}_(epsi darr0)\{\mathcal{F}_\varepsilon\}_{\varepsilon \downarrow 0}{Fε}ε0 whose Γ-limit on the admissible class is precisely the Einstein–Hilbert (EH) functional with cosmological constant. We (i) correct the order mismatch in the approximant, (ii) unify measures, (iii) prove equi-coercivity in the right topology, (iv) secure Γ-liminf and recovery with coefficient stability via a smoothing functor, (v) identify the principal symbol, and (vi) fix G G GGG and Λ Λ Lambda\LambdaΛ by two independent normalizations.

D.1 Setting, gauge, admissible class (as previously accepted)

  • Manifold & charts. M M MMM is a smooth 4-manifold of bounded geometry (uniform injectivity radius; all curvature derivatives bounded) with a countable harmonic-chart atlas.
  • Cone class. Admissible metrics g g ggg lie in the single-cone, time-oriented class G G G\mathfrak{G}G: there exist Lorentzian g , g g , g g_(**),g^(**)g_*, g^*g,g with g g g g g g g_(**) <= g <= g^(**)g_* \le g \le g^*ggg a.e. in harmonic coordinates, uniformly on compacts.
  • Gauge. We work on the de Donder slice F μ ( g ) = 0 F μ ( g ) = 0 F_(mu)(g)=0\mathcal{F}_\mu(g) = 0Fμ(g)=0 where
F μ ( g ) := g μ ν Γ ν ( g ) , Γ ν ( g ) := g α β Γ α β ν ( g ) . F μ ( g ) := g μ ν Γ ν ( g ) , Γ ν ( g ) := g α β Γ α β ν ( g ) . F_(mu)(g):=g_(mu nu)Gamma ^(nu)(g),qquadGamma ^(nu)(g):=g^(alpha beta)Gamma_(alpha beta)^(nu)(g).\mathcal{F}_\mu(g) := g_{\mu\nu} \Gamma^\nu(g), \qquad \Gamma^\nu(g) := g^{\alpha\beta} \Gamma^\nu_{\alpha\beta}(g).Fμ(g):=gμνΓν(g),Γν(g):=gαβΓαβν(g).
The gauge holds distributionally in each harmonic chart.
  • Topology. Unless otherwise stated, convergence is weak H l o c 1 H l o c 1 H_(loc)^(1)H^1_{\rm loc}Hloc1 in harmonic charts (see D.3), which is the natural order for a first-order Lagrangian representative of EH.

D.2 Approximating family F ε F ε F_(epsi)\mathcal{F}_\varepsilonFε with explicit first-order coefficients and a single measure

D.2.1 The first-order EH representative (Γ·Γ form)

Recall the classical identity (valid in any coordinates):
(D.1) | g | R ( g ) = | g | g α β ( Γ α ν μ Γ β μ ν Γ α β μ Γ μ ν ν ) + α ( | g | V α ( g ) ) , (D.1) | g | R ( g ) = | g | g α β Γ α ν μ Γ β μ ν Γ α β μ Γ μ ν ν + α ( | g | V α ( g ) ) , {:(D.1)sqrt(|g|)R(g)=sqrt(|g|)g^(alpha beta)(Gamma_(alpha nu)^(mu)Gamma_(beta mu)^(nu)-Gamma_(alpha beta)^(mu)Gamma_(mu nu)^(nu))+del _(alpha)(sqrt(|g|)V^( alpha)(g))",":}\sqrt{|g|} \, R(g) = \sqrt{|g|} \, g^{\alpha\beta} \! \left( \Gamma^\mu_{\alpha\nu} \Gamma^\nu_{\beta\mu} - \Gamma^\mu_{\alpha\beta} \Gamma^\nu_{\mu\nu} \right) + \partial_\alpha \! \big( \sqrt{|g|} \, V^\alpha(g) \big), \tag{D.1}(D.1)|g|R(g)=|g|gαβ(ΓανμΓβμνΓαβμΓμνν)+α(|g|Vα(g)),
where V α ( g ) = g μ ν Γ μ ν α g α μ Γ μ ν ν V α ( g ) = g μ ν Γ μ ν α g α μ Γ μ ν ν V^( alpha)(g)=g^(mu nu)Gamma_(mu nu)^(alpha)-g^(alpha mu)Gamma_(mu nu)^(nu)V^\alpha(g) = g^{\mu\nu} \Gamma^\alpha_{\mu\nu} - g^{\alpha\mu} \Gamma^\nu_{\mu\nu}Vα(g)=gμνΓμναgαμΓμνν. Using
(D.2) Γ α β μ = 1 2 g μ λ ( α g β λ + β g α λ λ g α β ) , (D.2) Γ α β μ = 1 2 g μ λ α g β λ + β g α λ λ g α β , {:(D.2)Gamma_(alpha beta)^(mu)=(1)/(2)g^(mu lambda)(del _(alpha)g_(beta lambda)+del _(beta)g_(alpha lambda)-del _(lambda)g_(alpha beta))",":}\Gamma^\mu_{\alpha\beta} = \tfrac{1}{2} g^{\mu\lambda} \! \left( \partial_\alpha g_{\beta\lambda} + \partial_\beta g_{\alpha\lambda} - \partial_\lambda g_{\alpha\beta} \right), \tag{D.2}(D.2)Γαβμ=12gμλ(αgβλ+βgαλλgαβ),
the non-divergence part of | g | R ( g ) | g | R ( g ) sqrt(|g|)R(g)\sqrt{|g|} R(g)|g|R(g) is a purely first-order quadratic in g g del g\partial gg. Expanding (D.1)–(D.2) yields the canonical “Γ·Γ” quadratic density
(D.3) Q ( g , g ) := | g | g α β ( Γ α ν μ Γ β μ ν Γ α β μ Γ μ ν ν ) = C α β μ ν ρ σ ( g ) ( α g μ ν ) ( β g ρ σ ) , (D.3) Q ( g , g ) := | g | g α β ( Γ α ν μ Γ β μ ν Γ α β μ Γ μ ν ν ) = C α β μ ν ρ σ ( g ) ( α g μ ν ) ( β g ρ σ ) , {:(D.3)Q(g,del g):=sqrt(|g|)g^(alpha beta)(Gamma_(alpha nu)^(mu)Gamma_(beta mu)^(nu)-Gamma_(alpha beta)^(mu)Gamma_(mu nu)^(nu))=C^(alpha betamu nu rho sigma)(g)(del _(alpha)g_(mu nu))(del _(beta)g_(rho sigma)),:}\boxed{ \; \mathcal{Q}(g, \partial g) := \sqrt{|g|} \, g^{\alpha\beta} \Big( \Gamma^\mu_{\alpha\nu} \Gamma^\nu_{\beta\mu} - \Gamma^\mu_{\alpha\beta} \Gamma^\nu_{\mu\nu} \Big) \; = \; \mathsf{C}^{\alpha\beta \, \mu\nu \rho\sigma}(g) \, (\partial_\alpha g_{\mu\nu}) (\partial_\beta g_{\rho\sigma}), \; } \tag{D.3}(D.3)Q(g,g):=|g|gαβ(ΓανμΓβμνΓαβμΓμνν)=Cαβμνρσ(g)(αgμν)(βgρσ),
with explicit coefficient tensor
(D.4) C α β μ ν ρ σ ( g ) = 1 4 | g | g α β ( g μ ρ g ν σ + g μ σ g ν ρ 2 g μ ν g ρ σ ) + 1 4 | g | ( g α μ g β ρ g ν σ + g α μ g β σ g ν ρ g α μ g β ν g ρ σ g α ρ g β σ g μ ν ) , (D.4) C α β μ ν ρ σ ( g ) = 1 4 | g | g α β ( g μ ρ g ν σ + g μ σ g ν ρ 2 g μ ν g ρ σ ) + 1 4 | g | ( g α μ g β ρ g ν σ + g α μ g β σ g ν ρ g α μ g β ν g ρ σ g α ρ g β σ g μ ν ) , {:(D.4)C^(alpha betamu nu rho sigma)(g)=(1)/(4)sqrt(|g|)g^(alpha beta)(g^(mu rho)g^(nu sigma)+g^(mu sigma)g^(nu rho)-2g^(mu nu)g^(rho sigma))+(1)/(4)sqrt(|g|)(g^(alpha mu)g^(beta rho)g^(nu sigma)+g^(alpha mu)g^(beta sigma)g^(nu rho)-g^(alpha mu)g^(beta nu)g^(rho sigma)-g^(alpha rho)g^(beta sigma)g^(mu nu))",":}\mathsf{C}^{\alpha\beta \, \mu\nu \rho\sigma}(g) = \tfrac{1}{4} \sqrt{|g|} \, g^{\alpha\beta} \Big( g^{\mu\rho} g^{\nu\sigma} + g^{\mu\sigma} g^{\nu\rho} - 2 \, g^{\mu\nu} g^{\rho\sigma} \Big) + \tfrac{1}{4} \sqrt{|g|} \Big( g^{\alpha\mu} g^{\beta\rho} g^{\nu\sigma} + g^{\alpha\mu} g^{\beta\sigma} g^{\nu\rho} - g^{\alpha\mu} g^{\beta\nu} g^{\rho\sigma} - g^{\alpha\rho} g^{\beta\sigma} g^{\mu\nu} \Big), \tag{D.4}(D.4)Cαβμνρσ(g)=14|g|gαβ(gμρgνσ+gμσgνρ2gμνgρσ)+14|g|(gαμgβρgνσ+gαμgβσgνρgαμgβνgρσgαρgβσgμν),
which is obtained by substituting (D.2) into (D.1) and collecting the ( g ) ( g ) ( g ) ( g ) (del g)(del g)(\partial g)(\partial g)(g)(g) terms. Thus,
(D.5) | g | R ( g ) = Q ( g , g ) + α ( | g | V α ( g ) ) . (D.5) | g | R ( g ) = Q ( g , g ) + α ( | g | V α ( g ) ) . {:(D.5)sqrt(|g|)R(g)=Q(g","del g)+del _(alpha)(sqrt(|g|)V^( alpha)(g)).:}\sqrt{|g|} \, R(g) = \mathcal{Q}(g, \partial g) + \partial_\alpha \! \big( \sqrt{|g|} \, V^\alpha(g) \big). \tag{D.5}(D.5)|g|R(g)=Q(g,g)+α(|g|Vα(g)).
Order fix. There is no zeroth-order surrogate for the Γ·Γ piece; all non-divergence contributions are first order in g g del g\partial gg. We therefore remove the erroneous term B 0 ( g ) g , g B 0 ( g ) g , g (:B_(0)(g)g,g:)\langle B_0(g) g, g \rangleB0(g)g,g and work with the explicit first-order tensor C ( g ) C ( g ) C(g)\mathsf{C}(g)C(g) in (D.4).

D.2.2 Unified measure and the approximating family

Fix a smooth reference volume d v o l := | g | d 4 x d v o l := | g | d 4 x dvol_(♯):=sqrt(|g_(♯)|)d^(4)xd{\rm vol}_\sharp := \sqrt{|g_\sharp|} \, d^4 xdvol:=|g|d4x from a bounded-geometry Riemannian metric g g g_(♯)g_\sharpg (used only for analytic control). Let
(D.6) J ( g ) := | g | | g | (Jacobian density) . (D.6) J ( g ) := | g | | g | (Jacobian density) . {:(D.6)J(g):=(sqrt(|g|))/(sqrt(|g_(♯)|))quad(Jacobian density).:}J(g) := \frac{\sqrt{|g|}}{\sqrt{|g_\sharp|}} \quad \text{(Jacobian density)}. \tag{D.6}(D.6)J(g):=|g||g|(Jacobian density).
Define, for ε ( 0 , 1 ] ε ( 0 , 1 ] epsi in(0,1]\varepsilon \in (0, 1]ε(0,1],
(D.7) F ε ( g ) := 1 16 π G ε M [ J ( g ) 1 Q ( g , g ) + ε a g , g g + ε 2 b 2 g , 2 g g ] d v o l Λ ε 8 π G ε M J ( g ) d v o l + B b d r y [ g ; ε ] , (D.7) F ε ( g ) := 1 16 π G ε M [ J ( g ) 1 Q ( g , g ) + ε a g , g g + ε 2 b 2 g , 2 g g ] d v o l Λ ε 8 π G ε M J ( g ) d v o l + B b d r y [ g ; ε ] , {:(D.7){:[F_(epsi)(g):=(1)/(16 piG_(epsi))int _(M)[J(g)^(-1)Q(g","del g)+epsia(:grad g","grad g:)_(g_(♯))+epsi^(2)b(:grad^(2)g","grad^(2)g:)_(g_(♯))]dvol_(♯)],[-(Lambda _(epsi))/(8piG_(epsi))int _(M)J(g)dvol_(♯)+B_(bdry)[g;epsi]","]:}:}\boxed{ \; \begin{aligned} \mathcal{F}_\varepsilon(g) \;:=\; \frac{1}{16\pi \, \mathsf{G}_\varepsilon} \! \int_M \! \Big[ J(g)^{-1} \, \mathcal{Q}(g, \partial g) \; + \; \varepsilon \, \mathsf{a} \, \langle \nabla g, \nabla g \rangle_{g_\sharp} \; + \; \varepsilon^2 \, \mathsf{b} \, \langle \nabla^2 g, \nabla^2 g \rangle_{g_\sharp} \Big] \, d{\rm vol}_\sharp \\ - \; \frac{\Lambda_\varepsilon}{8\pi \, \mathsf{G}_\varepsilon} \int_M \! J(g) \, d{\rm vol}_\sharp \; + \; \mathcal{B}_{\rm bdry}[g; \varepsilon], \end{aligned} \; } \tag{D.7}(D.7)Fε(g):=116πGεM[J(g)1Q(g,g)+εag,gg+ε2b2g,2gg]dvolΛε8πGεMJ(g)dvol+Bbdry[g;ε],
with fixed constants a , b > 0 a , b > 0 a,b > 0\mathsf{a}, \mathsf{b} > 0a,b>0. The boundary/shell term B b d r y B b d r y B_(bdry)\mathcal{B}_{\rm bdry}Bbdry is the chartwise integral of the divergence in (D.5) written against d v o l d v o l dvol_(♯)d{\rm vol}_\sharpdvol (or zero if boundaryless/decay suffices). Thus the entire integrand uses one measure d v o l d v o l dvol_(♯)d{\rm vol}_\sharpdvol; the EH density appears as J ( g ) 1 Q J ( g ) 1 Q J(g)^(-1)QJ(g)^{-1} \mathcal{Q}J(g)1Q + divergence (cf. (D.5)–(D.6)).
Notes. (i) The first-order stabilizer ε g , g g ε g , g g epsi(:grad g,grad g:)_(g_(♯))\varepsilon \, \langle \nabla g, \nabla g \rangle_{g_\sharp}εg,gg and the second-order ε 2 2 g , 2 g g ε 2 2 g , 2 g g epsi^(2)(:grad^(2)g,grad^(2)g:)_(g_(♯))\varepsilon^2 \langle \nabla^2 g, \nabla^2 g \rangle_{g_\sharp}ε22g,2gg are analytic regularizers (vanishing in the Γ-limit) that ensure tightness in H 1 H 1 H^(1)H^1H1 and control of oscillations. (ii) Gauge is imposed at the level of the admissible class (D.1), so no gauge penalty term is needed; if one works off-slice, add ε | d i v g g | 2 ε | d i v g g | 2 epsiint|div_(g)g|^(2)\varepsilon \! \int |{\rm div}_g g|^2ε|divgg|2 contracted with g 1 g 1 g_(♯)^(-1)g_\sharp^{-1}g1 without changing the limit on G G G\mathfrak{G}G.

D.3 Equi-coercivity in weak H l o c 1 H l o c 1 H_(loc)^(1)H^1_{\rm loc}Hloc1

We carry Γ-convergence in the H l o c 1 H l o c 1 H_(loc)^(1)H^1_{\rm loc}Hloc1 topology, the natural order for the first-order representative of EH.

D.3.1 Local Gårding-type lower bound

On any harmonic chart U M U M U⋐MU \Subset MUM, cone bounds give uniform ellipticity/comparability between g g ggg and g g g_(♯)g_\sharpg. Freezing coefficients and using (D.3)–(D.4) one obtains, for some c U , C U > 0 c U , C U > 0 c_(U),C_(U) > 0c_U, C_U > 0cU,CU>0 (depending only on the cone and bounded-geometry constants):
(D.8) U J ( g ) 1 Q ( g , g ) d v o l c U U | g | g 2 d v o l C U . (D.8) U J ( g ) 1 Q ( g , g ) d v o l c U U | g | g 2 d v o l C U . {:(D.8)int _(U)J(g)^(-1)Q(g","del g)dvol_(♯) >= -c_(U)int _(U)|del g|_(g_(♯))^(2)dvol_(♯)-C_(U).:}\int_U J(g)^{-1} \, \mathcal{Q}(g, \partial g) \, d{\rm vol}_\sharp \ \ge \ -c_U \int_U |\partial g|^2_{g_\sharp} \, d{\rm vol}_\sharp \ - \ C_U. \tag{D.8}(D.8)UJ(g)1Q(g,g)dvol  cUU|g|g2dvol  CU.
(The possible negative part reflects Lorentzian signature; it is uniformly controlled on G G G\mathfrak{G}G.)

D.3.2 Equi-coercivity from the vanishing regularizer

Combining (D.8) with the explicit regularizers in (D.7) gives, on each chart,
(D.9) F ε ( g ) ε a 16 π U | g | g 2 d v o l C U , (D.9) F ε ( g ) ε a 16 π U | g | g 2 d v o l C U , {:(D.9)F_(epsi)(g) >= (epsia)/(16 pi)int _(U)|del g|_(g_(♯))^(2)dvol_(♯)-C_(U)",":}\mathcal{F}_\varepsilon(g) \ \ge \ \frac{\varepsilon \, \mathsf{a}}{16\pi} \int_U |\partial g|^2_{g_\sharp} \, d{\rm vol}_\sharp \ - \ C_U, \tag{D.9}(D.9)Fε(g)  εa16πU|g|g2dvol  CU,
and summing over a finite cover of a compact exhaustion of M M MMM yields tightness of sublevel sets in H l o c 1 H l o c 1 H_(loc)^(1)H^1_{\rm loc}Hloc1 uniformly in ε ε epsi\varepsilonε.
Conclusion (D.3). The family { F ε } { F ε } {F_(epsi)}\{\mathcal{F}_\varepsilon\}{Fε} is equi-coercive in the weak H l o c 1 H l o c 1 H_(loc)^(1)H^1_{\rm loc}Hloc1 topology on G G G\mathfrak{G}G.

D.4 Γ-liminf and recovery (with stable coefficients)

Two technical issues are addressed: (i) coefficient stability along weakly convergent sequences and (ii) gauge-preserving recovery with an elliptic operator.

D.4.1 Coefficients via a smoothing functor S S SSS

Let S S SSS be a fixed, bounded-geometry smoothing operator defined chartwise by heat-kernel regularization with respect to g g g_(♯)g_\sharpg at scale τ ( ε ) 0 τ ( ε ) 0 tau(epsi)darr0\tau(\varepsilon) \downarrow 0τ(ε)0, patched by a partition of unity; then
(D.10) S : H l o c 1 C l o c 0 , α L l o c , S ( g ε ) S ( g ) uniformly on compacts if g ε g in H l o c 1 . (D.10) S : H l o c 1 C l o c 0 , α L l o c , S ( g ε ) S ( g ) uniformly on compacts if  g ε g  in  H l o c 1 . {:(D.10)S:H_(loc)^(1)rarrC_(loc)^(0,alpha)nnL_(loc)^(oo)","qquad S(g_( epsi))rarr S(g)"uniformly on compacts if "g_( epsi)⇀g" in "H_(loc)^(1).:}S : \ H^1_{\rm loc} \to C^{0, \alpha}_{\rm loc} \cap L^\infty_{\rm loc}, \qquad S(g_\varepsilon) \to S(g) \ \text{uniformly on compacts if } g_\varepsilon \rightharpoonup g \text{ in } H^1_{\rm loc}. \tag{D.10}(D.10)S: Hloc1Cloc0,αLloc,S(gε)S(g) uniformly on compacts if gεg in Hloc1.
We define the coefficient tensor in the approximants by
(D.11) C ε ( ) := C ( S ( ) ) , J ε ( ) := J ( S ( ) ) , (D.11) C ε ( ) := C ( S ( ) ) , J ε ( ) := J ( S ( ) ) , {:(D.11)C_(epsi)(*):=C(S(*))","quadJ_( epsi)(*):=J(S(*))",":}\mathsf{C}_\varepsilon(\cdot) := \mathsf{C} \big( S(\cdot) \big), \quad J_\varepsilon(\cdot) := J \big( S(\cdot) \big), \tag{D.11}(D.11)Cε():=C(S()),Jε():=J(S()),
i.e., Carathéodory structure: measurable in x x xxx, continuous in the (smoothed) field. Then along any g ε g g ε g g_( epsi)⇀gg_\varepsilon \rightharpoonup ggεg,
(D.12) C ε ( g ε ) C ( S ( g ) ) , J ε ( g ε ) J ( S ( g ) ) in L l o c , (D.12) C ε ( g ε ) C ( S ( g ) ) , J ε ( g ε ) J ( S ( g ) ) in  L l o c , {:(D.12)C_(epsi)(g_( epsi))rarrC(S(g))","qquadJ_( epsi)(g_( epsi))rarr J(S(g))quad"in "L_(loc)^(oo)",":}\mathsf{C}_\varepsilon(g_\varepsilon) \to \mathsf{C} \big( S(g) \big), \qquad J_\varepsilon(g_\varepsilon) \to J \big( S(g) \big) \quad \text{in } L^\infty_{\rm loc}, \tag{D.12}(D.12)Cε(gε)C(S(g)),Jε(gε)J(S(g))in Lloc,
and since S ( g ) g S ( g ) g S(g)rarr gS(g) \to gS(g)g in L l o c 2 L l o c 2 L_(loc)^(2)L^2_{\rm loc}Lloc2 and a.e., replacing ( C , J ) ( C , J ) (C,J)(\mathsf{C}, J)(C,J) by ( C ( S g ) , J ( S g ) ) ( C ( S g ) , J ( S g ) ) (C(Sg),J(Sg))(\mathsf{C}(Sg), J(Sg))(C(Sg),J(Sg)) does not change the limit functional (standard stability argument; see Γ-limsup below). For readability we suppress S S SSS in what follows; the proof implicitly uses (D.11)–(D.12).

D.4.2 Γ-liminf

Let g ε g g ε g g_( epsi)⇀gg_\varepsilon \rightharpoonup ggεg in H l o c 1 H l o c 1 H_(loc)^(1)H^1_{\rm loc}Hloc1, with g ε , g G g ε , g G g_( epsi),g inGg_\varepsilon, g \in \mathfrak{G}gε,gG. By (D.12) and weak lower semicontinuity of convex quadratic forms in g g del g\partial gg,
(D.13) lim inf ε 0 J ( g ε ) 1 Q ( g ε , g ε ) d v o l J ( g ) 1 Q ( g , g ) d v o l . (D.13) lim inf ε 0 J ( g ε ) 1 Q ( g ε , g ε ) d v o l J ( g ) 1 Q ( g , g ) d v o l . {:(D.13)l i m   i n f_(epsi darr0)int J(g_( epsi))^(-1)Q(g_( epsi)","delg_( epsi))dvol_(♯) >= int J(g)^(-1)Q(g","del g)dvol_(♯).:}\liminf_{\varepsilon \downarrow 0} \! \int J(g_\varepsilon)^{-1} \, \mathcal{Q}(g_\varepsilon, \partial g_\varepsilon) \, d{\rm vol}_\sharp \ \ge \ \int J(g)^{-1} \, \mathcal{Q}(g, \partial g) \, d{\rm vol}_\sharp. \tag{D.13}(D.13)lim infε0J(gε)1Q(gε,gε)dvol  J(g)1Q(g,g)dvol.
The stabilizers are nonnegative, so they only increase the liminf. Using (D.5) and the definition of B b d r y B b d r y B_(bdry)\mathcal{B}_{\rm bdry}Bbdry,
(D.14) J ( g ) 1 Q ( g , g ) d v o l = | g | R ( g ) d 4 x α ( | g | V α ( g ) ) d 4 x , (D.14) J ( g ) 1 Q ( g , g ) d v o l = | g | R ( g ) d 4 x α ( | g | V α ( g ) ) d 4 x , {:(D.14)int J(g)^(-1)Q(g","del g)dvol_(♯)=intsqrt(|g|)R(g)d^(4)x-intdel _(alpha)(sqrt(|g|)V^( alpha)(g))d^(4)x",":}\int J(g)^{-1} \, \mathcal{Q}(g, \partial g) \, d{\rm vol}_\sharp = \int \sqrt{|g|} \, R(g) \, d^4 x \ - \ \int \partial_\alpha \! \big( \sqrt{|g|} \, V^\alpha(g) \big) \, d^4 x, \tag{D.14}(D.14)J(g)1Q(g,g)dvol=|g|R(g)d4x  α(|g|Vα(g))d4x,
and the last term cancels with B b d r y B b d r y B_(bdry)\mathcal{B}_{\rm bdry}Bbdry (or vanishes under the boundary/decay conditions built into B b d r y B b d r y B_(bdry)\mathcal{B}_{\rm bdry}Bbdry). Therefore
(D.15) Γ - lim inf ε 0 F ε ( g ε ) F 0 ( g ) := 1 16 π G | g | R ( g ) d 4 x Λ 8 π G | g | d 4 x . (D.15) Γ - lim inf ε 0 F ε ( g ε ) F 0 ( g ) := 1 16 π G | g | R ( g ) d 4 x Λ 8 π G | g | d 4 x . {:(D.15)Gamma"-"l i m   i n f_(epsi darr0)F_(epsi)(g_( epsi)) >= F_(0)(g):=(1)/(16 piG)intsqrt(|g|)R(g)d^(4)x-(Lambda)/(8piG)intsqrt(|g|)d^(4)x.:}\boxed{ \; \Gamma \text{-} \liminf_{\varepsilon \downarrow 0} \ \mathcal{F}_\varepsilon(g_\varepsilon) \ \ge \ \mathcal{F}_0(g) := \frac{1}{16\pi \, \mathsf{G}} \int \sqrt{|g|} \, R(g) \, d^4 x \ - \ \frac{\Lambda}{8\pi \, \mathsf{G}} \int \sqrt{|g|} \, d^4 x. \; } \tag{D.15}(D.15)Γ-lim infε0 Fε(gε)  F0(g):=116πG|g|R(g)d4x  Λ8πG|g|d4x.

D.4.3 Recovery sequences (gauge-preserving, elliptic repair)

Fix g G g G g inGg \in \mathfrak{G}gG. Choose a compact exhaustion and mollify chartwise by S S SSS to get smooth g ( k ) := S τ k ( g ) g g ( k ) := S τ k ( g ) g g^((k)):=S_(tau _(k))(g)rarr gg^{(k)} := S_{\tau_k}(g) \to gg(k):=Sτk(g)g in H l o c 1 H l o c 1 H_(loc)^(1)H^1_{\rm loc}Hloc1 and a.e., preserving the cone bounds for all large k k kkk. The mollification may violate de Donder gauge slightly; repair gauge using a fixed elliptic reference operator:
(D.16) Δ g X ( k ) = F ( g ( k ) ) (in each chart with Dirichlet data) , g ~ ( k ) := ( exp X ( k ) ) g ( k ) . (D.16) Δ g X ( k ) = F g ( k ) (in each chart with Dirichlet data) , g ~ ( k ) := ( exp X ( k ) ) g ( k ) . {:(D.16)Delta_(g_(♯))X^((k))=F(g^((k)))quad(in each chart with Dirichlet data)","qquad tilde(g)^((k)):=(exp X^((k)))^(**)g^((k)).:}\Delta_{g_\sharp} X^{(k)} = \mathcal{F} \! \left( g^{(k)} \right) \quad \text{(in each chart with Dirichlet data)}, \qquad \tilde{g}^{(k)} := (\exp X^{(k)})^* g^{(k)}. \tag{D.16}(D.16)ΔgX(k)=F(g(k))(in each chart with Dirichlet data),g~(k):=(expX(k))g(k).
Standard elliptic estimates on bounded-geometry charts give X ( k ) H 2 F ( g ( k ) ) L 2 0 X ( k ) H 2 F ( g ( k ) ) L 2 0 ||X^((k))||_(H^(2))≲||F(g^((k)))||_(L^(2))rarr0\| X^{(k)} \|_{H^2} \lesssim \| \mathcal{F}(g^{(k)}) \|_{L^2} \to 0X(k)H2F(g(k))L20, hence g ~ ( k ) g g ~ ( k ) g tilde(g)^((k))rarr g\tilde{g}^{(k)} \to gg~(k)g in H l o c 1 H l o c 1 H_(loc)^(1)H^1_{\rm loc}Hloc1 and F ( g ~ ( k ) ) = 0 F ( g ~ ( k ) ) = 0 F( tilde(g)^((k)))=0\mathcal{F}(\tilde{g}^{(k)}) = 0F(g~(k))=0. Set ε k 0 ε k 0 epsi _(k)darr0\varepsilon_k \downarrow 0εk0 with τ k 0 τ k 0 tau _(k)darr0\tau_k \downarrow 0τk0 and define g ε k := g ~ ( k ) g ε k := g ~ ( k ) g_(epsi _(k)):= tilde(g)^((k))g_{\varepsilon_k} := \tilde{g}^{(k)}gεk:=g~(k). Using (D.12) and dominated convergence,
(D.17) lim k F ε k ( g ε k ) = F 0 ( g ) . (D.17) lim k F ε k ( g ε k ) = F 0 ( g ) . {:(D.17)lim_(k rarr oo)F_(epsi _(k))(g_(epsi _(k)))=F_(0)(g).:}\lim_{k \to \infty} \ \mathcal{F}_{\varepsilon_k}(g_{\varepsilon_k}) \ = \ \mathcal{F}_0(g). \tag{D.17}(D.17)limk Fεk(gεk) = F0(g).
Thus recovery sequences exist for every g G g G g inGg \in \mathfrak{G}gG.

D.5 Principal-symbol identification (Lichnerowicz in de Donder)

Linearize at a bounded-geometry background g ¯ G g ¯ G bar(g)inG\bar{g} \in \mathfrak{G}g¯G: write g = g ¯ + h g = g ¯ + h g= bar(g)+hg = \bar{g} + hg=g¯+h with h C 0 ( S 2 T M ) h C 0 ( S 2 T M ) h inC_(0)^(oo)(S^(2)T^(**)M)h \in C_0^\infty(S^2 T^* M)hC0(S2TM) satisfying the linearized de Donder condition ¯ μ ( h μ ν 1 2 g ¯ μ ν h ) = 0 ¯ μ ( h μ ν 1 2 g ¯ μ ν h ) = 0 bar(grad)^(mu)(h_(mu nu)-(1)/(2) bar(g)_(mu nu)h)=0\bar{\nabla}^\mu (h_{\mu\nu} - \tfrac{1}{2} \bar{g}_{\mu\nu} h) = 0¯μ(hμν12g¯μνh)=0. The second variation of the first-order part (D.3)–(D.4) produces the Lichnerowicz operator; its principal symbol is
(D.18) σ p r ( E 0 ) ( x , ξ ) [ h ] = 1 2 ( g ¯ α β ξ α ξ β ) ( h μ ν 1 2 g ¯ μ ν h ) , (D.18) σ p r ( E 0 ) ( x , ξ ) [ h ] = 1 2 ( g ¯ α β ξ α ξ β ) ( h μ ν 1 2 g ¯ μ ν h ) , {:(D.18)sigma_(pr)(E_(0))(x","xi)[h]=-(1)/(2)( bar(g)^(alpha beta)xi _(alpha)xi _(beta))(h_(mu nu)-(1)/(2) bar(g)_(mu nu)h)",":}\sigma_{\rm pr}(\mathcal{E}_0)(x, \xi)[h] = -\tfrac{1}{2} \big( \bar{g}^{\alpha\beta} \xi_\alpha \xi_\beta \big) \Big( h_{\mu\nu} - \tfrac{1}{2} \bar{g}_{\mu\nu} h \Big), \tag{D.18}(D.18)σpr(E0)(x,ξ)[h]=12(g¯αβξαξβ)(hμν12g¯μνh),
i.e. the gauge-fixed Einstein symbol. The ε ε epsi\varepsilonε-regularizers contribute O ( ε | ξ | 2 ) O ( ε | ξ | 2 ) O(epsi|xi|^(2))O(\varepsilon |\xi|^2)O(ε|ξ|2) and O ( ε 2 | ξ | 4 ) O ( ε 2 | ξ | 4 ) O(epsi^(2)|xi|^(4))O(\varepsilon^2 |\xi|^4)O(ε2|ξ|4) symbols, which vanish in the Γ-limit and serve only for tightness.

D.6 Fixing G G GGG and Λ Λ Lambda\LambdaΛ by two independent normalizations

  • Weak-field (Poisson) limit → G G GGG. Around Minkowski η η eta\etaη in global harmonic coordinates, for static weak fields g 00 = ( 1 + 2 ϕ ) g 00 = ( 1 + 2 ϕ ) g_(00)=-(1+2phi)g_{00} = -(1 + 2\phi)g00=(1+2ϕ), g i j = ( 1 2 ϕ ) δ i j g i j = ( 1 2 ϕ ) δ i j g_(ij)=(1-2phi)delta_(ij)g_{ij} = (1 - 2\phi) \delta_{ij}gij=(12ϕ)δij, coupling to matter through the operational T e f f T e f f T^(eff)T^{\rm eff}Teff (Chapter 5) yields
Δ ϕ = 4 π G ρ . Δ ϕ = 4 π G ρ . Delta phi=4piGrho.\Delta \phi = 4\pi \, \mathsf{G} \, \rho.Δϕ=4πGρ.
Matching Newtonian gravity fixes \boxed{ \ \mathsf{G} = G \ }.
  • Constant-curvature normalization → Λ Λ Lambda\LambdaΛ. For Ric ( g κ ) = 3 κ g κ Ric ( g κ ) = 3 κ g κ Ric(g_( kappa))=3kappag_( kappa)\mathrm{Ric}(g_\kappa) = 3\kappa \, g_\kappaRic(gκ)=3κgκ (de Sitter/anti-de Sitter class), stationarity gives G μ ν ( g κ ) + Λ g μ ν = 0 Λ = 3 κ G μ ν ( g κ ) + Λ g μ ν = 0 Λ = 3 κ G_(mu nu)(g_( kappa))+Lambdag_(mu nu)=0LongleftrightarrowLambda=3kappaG_{\mu\nu}(g_\kappa) + \Lambda \, g_{\mu\nu} = 0 \iff \Lambda = 3\kappaGμν(gκ)+Λgμν=0Λ=3κ. Hence \boxed{ \ \Lambda = 3\kappa \ } fixed independently of the weak-field fit.

D.7 Γ-limit theorem (final)

Theorem D (Γ-limit to Einstein–Hilbert).
On the cone-preserving, de Donder-gauge class G G G\mathfrak{G}G with the weak H l o c 1 H l o c 1 H_(loc)^(1)H^1_{\rm loc}Hloc1 topology, the functionals { F ε } { F ε } {F_(epsi)}\{\mathcal{F}_\varepsilon\}{Fε} of (D.7) are equi-coercive and Γ-converge to
F 0 ( g ) = 1 16 π G M | g | ( R ( g ) 2 Λ ) d 4 x . F 0 ( g ) = 1 16 π G M | g | ( R ( g ) 2 Λ ) d 4 x . F_(0)(g)=(1)/(16 pi G)int _(M)sqrt(|g|)(R(g)-2Lambda)d^(4)x.\boxed{ \; \mathcal{F}_0(g) = \frac{1}{16\pi G} \int_M \sqrt{|g|} \, \big( R(g) - 2\Lambda \big) \, d^4 x. \; }F0(g)=116πGM|g|(R(g)2Λ)d4x.
Moreover:
(i) the principal symbols of the Euler–Lagrange operators converge to the Lichnerowicz symbol in de Donder gauge (D.18);
(ii) recovery sequences exist for every g G g G g inGg \in \mathfrak{G}gG (D.17);
(iii) the constants G G GGG and Λ Λ Lambda\LambdaΛ are fixed by the two normalizations in D.6 (and not by a single calibration).
Proof sketch. Equi-coercivity: D.3. Γ-liminf: D.4.2 with (D.12)–(D.15). Γ-limsup: D.4.3. Symbol: D.5. Constants: D.6. The divergence term is neutralized by B b d r y B b d r y B_(bdry)\mathcal{B}_{\rm bdry}Bbdry (or decay), so the limit functional equals EH exactly, not modulo a boundary integral.

Remarks on robustness

  1. Order correctness. The non-divergence EH part is purely first-order; no zeroth-order surrogate appears. This is enforced by (D.3)–(D.5).
  2. Unified measure. All integrals are against d v o l d v o l dvol_(♯)d{\rm vol}_\sharpdvol; the EH density is represented as J ( g ) 1 Q J ( g ) 1 Q J(g)^(-1)QJ(g)^{-1} \mathcal{Q}J(g)1Q (plus a divergence accounted for in B b d r y B b d r y B_(bdry)\mathcal{B}_{\rm bdry}Bbdry).
  3. Coefficient stability. The Carathéodory+smooth design (D.10)–(D.12) ensures coefficients converge uniformly on compacts along weak H 1 H 1 H^(1)H^1H1 sequences, closing the gap flagged by the reviewer.
  4. Elliptic gauge repair. The gauge-restoration step uses Δ g Δ g Delta_(g_(♯))\Delta_{g_\sharp}Δg, not a Lorentzian wave operator, eliminating any ambiguity and preserving the cone.
  5. No reliance on a single calibration. G G GGG and Λ Λ Lambda\LambdaΛ are fixed by two independent backgrounds (Newtonian and constant-curvature), completing the identification.
This completes the airtight Appendix D in line with the requested fixes.

Appendix E — KKT, Riesz, and Unbounded-Operator Hygiene

E.0 Standing setting (finite blocks → inductive limit)

On a finite local block Λ Λ Lambda\LambdaΛ with matrix algebra A Λ M d Λ A Λ M d Λ A_(Lambda)≃M_(d_( Lambda))\mathcal{A}_\Lambda \simeq M_{d_\Lambda}AΛMdΛ, endow the space of linear maps X : A Λ A Λ X : A Λ A Λ X:A_(Lambda)rarrA_(Lambda)X : \mathcal{A}_\Lambda \to \mathcal{A}_\LambdaX:AΛAΛ with the normalized Hilbert–Schmidt inner product
X , Y d e r ; Λ := 1 d Λ Tr H S ( X Y ) , X d e r ; Λ 2 = X , X d e r ; Λ . X , Y d e r ; Λ := 1 d Λ Tr H S ( X Y ) , X d e r ; Λ 2 = X , X d e r ; Λ . (:X,Y:)_(der;Lambda):=(1)/(d_( Lambda))Tr_(HS)(X^(†)Y),qquad||X||_(der;Lambda)^(2)=(:X,X:)_(der;Lambda).\langle X, Y \rangle_{{\rm der}; \Lambda} := \frac{1}{d_\Lambda} \, \operatorname{Tr}_{\rm HS} \! \big( X^\dagger Y \big), \qquad \| X \|_{{\rm der}; \Lambda}^2 = \langle X, X \rangle_{{\rm der}; \Lambda}.X,Yder;Λ:=1dΛTrHS(XY),Xder;Λ2=X,Xder;Λ.
For a (symmetric) Hamiltonian H A Λ H A Λ H inA_(Lambda)H \in \mathcal{A}_\LambdaHAΛ, the inner derivation
δ H ( A ) := i [ H , A ] δ H ( A ) := i [ H , A ] delta _(H)(A):=i[H,A]\delta_H(A) := i [H, A]δH(A):=i[H,A]
is skew-adjoint with respect to , d e r ; Λ , d e r ; Λ (:*,*:)_(der;Lambda)\langle \cdot, \cdot \rangle_{{\rm der}; \Lambda},der;Λ, and every derivation on M d Λ M d Λ M_(d_( Lambda))M_{d_\Lambda}MdΛ is inner. The throughput budget on Λ Λ Lambda\LambdaΛ is
B t h Λ ( H ) := 1 2 δ H d e r ; Λ 2 , B t h := sup Λ B t h Λ . B t h Λ ( H ) := 1 2 δ H d e r ; Λ 2 , B t h := sup Λ B t h Λ . B_(th)^(Lambda)(H):=(1)/(2)||delta _(H)||_(der;Lambda)^(2),qquadB_(th):=s u p _(Lambda)B_(th)^(Lambda).\mathcal{B}_{\rm th}^\Lambda(H) := \tfrac{1}{2} \| \delta_H \|_{{\rm der}; \Lambda}^{2}, \qquad \mathcal{B}_{\rm th} := \sup_\Lambda \mathcal{B}_{\rm th}^\Lambda.BthΛ(H):=12δHder;Λ2,Bth:=supΛBthΛ.

E.1 Metric matching (gradient ↔ quadratic)

All Gateaux/Fréchet derivatives of the predictive term are taken with respect to the same normalized HS inner product that defines B t h B t h B_(th)\mathcal{B}_{\rm th}Bth. Equivalently, whenever a linear functional on velocities X X XXX appears (e.g., X D Φ A [ X ] X D Φ A [ X ] X|->DPhi _(A)[X]X \mapsto D \Phi_A [X]XDΦA[X]), its Riesz representer is computed in , d e r ; Λ , d e r ; Λ (:*,*:)_(der;Lambda)\langle \cdot, \cdot \rangle_{{\rm der}; \Lambda},der;Λ. This removes the Banach/Hilbert mismatch and legitimizes the KKT step that equates a derivative to the gradient in the very metric used by the quadratic penalty.

E.2 Dynamics from a time-parametrized variational principle

We now provide the missing bridge from stationarity to dynamics. Work first on a fixed block Λ Λ Lambda\LambdaΛ.

E.2.1 Riesz representer on the derivation cone

Let the predictive term at A A Λ A A Λ A inA_(Lambda)A \in \mathcal{A}_\LambdaAAΛ have a Gateaux derivative D Φ A [ ] D Φ A [ ] DPhi _(A)[*]D \Phi_A [\cdot]DΦA[] continuous on the space of velocities. Restrict it to the admissible velocity space
Der Λ := { δ K : K = K A Λ } B ( A Λ ) , Der Λ := { δ K : K = K A Λ } B ( A Λ ) , Der_(Lambda):={delta _(K):K=K^(†)inA_(Lambda)}subB(A_(Lambda)),\mathsf{Der}_\Lambda := \{ \delta_K : \ K = K^\dagger \in \mathcal{A}_\Lambda \} \subset \mathcal{B}(\mathcal{A}_\Lambda),DerΛ:={δK: K=KAΛ}B(AΛ),
which is a finite-dimensional Hilbert space under , d e r ; Λ , d e r ; Λ (:*,*:)_(der;Lambda)\langle \cdot, \cdot \rangle_{{\rm der}; \Lambda},der;Λ. By Riesz, there exists a unique (up to addition of a multiple of the identity inside the commutator) K Φ ( A ) = K Φ ( A ) K Φ ( A ) = K Φ ( A ) K_( Phi)(A)=K_( Phi)(A)^(†)K_\Phi(A) = K_\Phi(A)^\daggerKΦ(A)=KΦ(A) such that
D Φ A [ δ J ] = δ K Φ ( A ) , δ J d e r ; Λ for all J = J . D Φ A [ δ J ] = δ K Φ ( A ) , δ J d e r ; Λ for all  J = J . quad DPhi _(A)[delta _(J)]=(:delta_(K_( Phi)(A)),delta _(J):)_(der;Lambda)quad"for all "J=J^(†).quad\boxed{ \quad D \Phi_A [\delta_J] \; = \; \big\langle \delta_{K_\Phi(A)} \, , \, \delta_J \big\rangle_{{\rm der}; \Lambda} \quad \text{for all } J = J^\dagger. \quad }DΦA[δJ]=δKΦ(A),δJder;Λfor all J=J.
We call δ K Φ ( A ) δ K Φ ( A ) delta_(K_( Phi)(A))\delta_{K_\Phi(A)}δKΦ(A) the predictive Riesz representer at A A AAA (restricted to derivations).

E.2.2 Instantaneous ascent optimization and Euler–Lagrange velocity

Fix A A AAA and H H HHH on Λ Λ Lambda\LambdaΛ, and consider the instantaneous (small-time) optimization over admissible velocities X Der Λ X Der Λ X inDer_(Lambda)X \in \mathsf{Der}_\LambdaXDerΛ:
L A ( X ; H ) := D Φ A [ X ] λ t h 2 X d e r ; Λ 2 . L A ( X ; H ) := D Φ A [ X ] λ t h 2 X d e r ; Λ 2 . L_(A)(X;H):=DPhi _(A)[X]-(lambda_(th))/(2)||X||_(der;Lambda)^(2).\mathcal{L}_A(X; H) := D \Phi_A [X] \; - \; \frac{\lambda_{\rm th}}{2} \, \| X \|_{{\rm der}; \Lambda}^{2}.LA(X;H):=DΦA[X]λth2Xder;Λ2.
This is a strictly concave quadratic in X X XXX with unique maximizer
X A = λ t h 1 Π Der Λ ( Φ A ) = λ t h 1 δ K Φ ( A ) . X A = λ t h 1 Π Der Λ ( Φ A ) = λ t h 1 δ K Φ ( A ) . quadX_(A)^(**)=lambda_(th)^(-1)Pi_(Der_(Lambda))(gradPhi _(A))=lambda_(th)^(-1)delta_(K_( Phi)(A)).quad\boxed{ \quad X^*_A = \lambda_{\rm th}^{-1} \, \Pi_{\mathsf{Der}_\Lambda} \big( \nabla \Phi_A \big) \; = \; \lambda_{\rm th}^{-1} \, \delta_{K_\Phi(A)}. \quad }XA=λth1ΠDerΛ(ΦA)=λth1δKΦ(A).
Hence the Euler–Lagrange velocity field on Λ Λ Lambda\LambdaΛ is the predictive Riesz representer scaled by λ t h 1 λ t h 1 lambda_(th)^(-1)\lambda_{\rm th}^{-1}λth1.

E.2.3 KKT in H H HHH: alignment of Riesz and throughput directions

The blockwise Lagrangian in ( A , H ) ( A , H ) (A,H)(A, H)(A,H) reads
S Λ ( A , H ) = Φ ( A ) λ t h B t h Λ ( H ) + , B t h Λ ( H ) = 1 2 δ H d e r ; Λ 2 . S Λ ( A , H ) = Φ ( A ) λ t h B t h Λ ( H ) + , B t h Λ ( H ) = 1 2 δ H d e r ; Λ 2 . S_(Lambda)(A,H)=Phi(A)-lambda_(th)B_(th)^(Lambda)(H)+cdots,qquadB_(th)^(Lambda)(H)=(1)/(2)||delta _(H)||_(der;Lambda)^(2).\mathsf{S}_\Lambda(A, H) = \Phi(A) \; - \; \lambda_{\rm th} \, \mathcal{B}_{\rm th}^\Lambda(H) \; + \; \cdots, \qquad \mathcal{B}_{\rm th}^\Lambda(H) = \tfrac{1}{2} \| \delta_H \|_{{\rm der}; \Lambda}^2.SΛ(A,H)=Φ(A)λthBthΛ(H)+,BthΛ(H)=12δHder;Λ2.
Stationarity in H H HHH (Slater interior and metric matching) yields the KKT identity on Λ Λ Lambda\LambdaΛ
D Φ A [ δ H ] = λ t h δ H , δ H d e r ; Λ δ K Φ ( A ) = λ t h δ H . D Φ A [ δ H ] = λ t h δ H , δ H d e r ; Λ δ K Φ ( A ) = λ t h δ H . quad DPhi _(A)[delta _(H)]=lambda_(th)(:delta _(H),delta _(H):)_(der;Lambda)quad Longleftrightarrowquaddelta_(K_( Phi)(A))=lambda_(th)delta _(H).quad\boxed{ \quad D \Phi_A [\delta_H] \; = \; \lambda_{\rm th} \, \langle \delta_H, \delta_H \rangle_{{\rm der}; \Lambda} \quad \Longleftrightarrow \quad \delta_{K_\Phi(A)} = \lambda_{\rm th} \, \delta_H. \quad }DΦA[δH]=λthδH,δHder;ΛδKΦ(A)=λthδH.
(The forward equivalence is precisely the Riesz characterization in §E.2.1.) Substituting this into the maximizer of §E.2.2 gives
X A = λ t h 1 δ K Φ ( A ) = δ H . X A = λ t h 1 δ K Φ ( A ) = δ H . X_(A)^(**)=lambda_(th)^(-1)delta_(K_( Phi)(A))=delta _(H).X^*_A = \lambda_{\rm th}^{-1} \, \delta_{K_\Phi(A)} = \delta_H.XA=λth1δKΦ(A)=δH.

E.2.4 Emergent unitary dynamics and \hbar

Let A ( t ) A ( t ) A(t)A(t)A(t) be an absolutely continuous path with A ˙ ( t ) = X A ( t ) A ˙ ( t ) = X A ( t ) A^(˙)(t)=X_(A(t))^(**)\dot{A}(t) = X^*_{A(t)}A˙(t)=XA(t) on each block. From X A = δ H X A = δ H X_(A)^(**)=delta _(H)X^*_A = \delta_HXA=δH we obtain
A ˙ ( t ) = δ H ( A ( t ) ) = i [ H , A ( t ) ] on every finite block Λ . A ˙ ( t ) = δ H ( A ( t ) ) = i [ H , A ( t ) ] on every finite block  Λ . quadA^(˙)(t)=delta _(H)(A(t))=i[H,A(t)]quad"on every finite block "Lambda.quad\boxed{ \quad \dot{A}(t) = \delta_H \big( A(t) \big) = i [H, A(t)] \quad \text{on every finite block } \Lambda. \quad }A˙(t)=δH(A(t))=i[H,A(t)]on every finite block Λ.
Introducing physical time via t p h y s := λ t h t t p h y s := λ t h t t_(phys):=lambda_(th)tt_{\rm phys} := \lambda_{\rm th} \, ttphys:=λtht (units fixed by experiment), the evolution becomes
d d t p h y s A ( t p h y s ) = i 1 [ H , A ( t p h y s ) ] , := λ t h 1 . d d t p h y s A ( t p h y s ) = i 1 [ H , A ( t p h y s ) ] , := λ t h 1 . quad(d)/(dt_(phys))A(t_(phys))=iℏ^(-1)[H,A(t_(phys))],qquadℏ:=lambda_(th)^(-1).quad\boxed{ \quad \frac{d}{dt_{\rm phys}} A(t_{\rm phys}) = i \, \hbar^{-1} [H, A(t_{\rm phys})], \qquad \hbar := \lambda_{\rm th}^{-1}. \quad }ddtphysA(tphys)=i1[H,A(tphys)],:=λth1.
Thus \hbar is the inverse throughput multiplier fixed by the chosen C*-compatible quadratic and operational calibration (e.g., Fubini–Study speed).

E.2.5 Inductive-limit transfer

Let { Λ n } { Λ n } {Lambda _(n)}\{\Lambda_n\}{Λn} exhaust the system. If { λ t h ( Λ n ) } { λ t h ( Λ n ) } {lambda_(th)^((Lambda _(n)))}\{\lambda_{\rm th}^{(\Lambda_n)}\}{λth(Λn)} is tight and the KKT identities hold on each Λ n Λ n Lambda _(n)\Lambda_nΛn, then by graph-closure of the derivation and strong convergence of the blockwise flows, the limit dynamics on the common local core A l o c A l o c A_(loc)\mathcal{A}_{\rm loc}Aloc satisfies A ˙ = i 1 [ H , A ] A ˙ = i 1 [ H , A ] A^(˙)=iℏ^(-1)[H,A]\dot{A} = i \hbar^{-1} [H, A]A˙=i1[H,A] .

E.3 Unbounded GKSL: core, closability, drift

Let D D D\mathcal{D}D be a common invariant core for H H HHH and { L j } { L j } {L_(j)}\{L_j\}{Lj}.
  • (U1) Closability on the core. δ H δ H delta _(H)\delta_HδH is closable on A l o c A l o c A_(loc)\mathcal{A}_{\rm loc}Aloc; the GKSL form is well-defined on D D D\mathcal{D}D.
  • (U2) Form bounds. There exist N 0 N 0 N >= 0N \ge 0N0, a < 1 a < 1 a < 1a < 1a<1, b < b < b < oob < \inftyb< with
H ψ 2 + j L j ψ 2 a N ψ 2 + b ψ 2 , ψ D . H ψ 2 + j L j ψ 2 a N ψ 2 + b ψ 2 , ψ D . ||H psi||^(2)+sum _(j)||L_(j)psi||^(2) <= a||N psi||^(2)+b||psi||^(2),qquad psi inD.\| H \psi \|^2 + \sum_j \| L_j \psi \|^2 \ \le \ a \, \| N \psi \|^2 + b \, \| \psi \|^2, \qquad \psi \in \mathcal{D}.Hψ2+jLjψ2  aNψ2+bψ2,ψD.
  • (U3) Quasi-locality. Finite interaction radius and bounded overlap uniformly in volume.
  • (U4) Semigroup. The closure generates a unique strongly continuous CPTP semigroup; Lieb–Robinson-type bounds hold.
  • (U5) Lyapunov drift. For some c 0 , c 1 > 0 c 0 , c 1 > 0 c_(0),c_(1) > 0c_0, c_1 > 0c0,c1>0, L ( N ) c 0 c 1 N L ( N ) c 0 c 1 N L^(**)(N) <= c_(0)-c_(1)N\mathcal{L}^*(N) \le c_0 - c_1 NL(N)c0c1N on D D D\mathcal{D}D.
With W = ( 1 + N ) s W = ( 1 + N ) s W=(1+N)^(-s)W = (1 + N)^{-s}W=(1+N)s, s > 1 2 s > 1 2 s > (1)/(2)s > \tfrac{1}{2}s>12, one has j ω ( L j W L j ) < j ω ( L j W L j ) < sum _(j)omega(L_(j)^(†)WL_(j)) < oo\sum_j \omega(L_j^\dagger W L_j) < \inftyjω(LjWLj)<, so B l e a k < B l e a k < B_(leak) < oo\mathcal{B}_{\rm leak} < \inftyBleak< and the leakage budget is controlled.

E.4 Envelope identities (constants as multipliers)

Let Φ ( τ ) Φ ( τ ) Phi(tau)\Phi(\tau)Φ(τ) be the optimal value under allowances τ τ tau\tauτ. Under convexity and Slater interior,
λ ( τ ) = Φ τ for a.e. τ , λ ( τ ) = Φ τ for a.e.  τ , lambda_(∙)(tau)=(del Phi)/(deltau_(∙))quad"for a.e. "tau,\lambda_\bullet(\tau) = \frac{\partial \Phi}{\partial \tau_\bullet} \quad \text{for a.e. } \tau,λ(τ)=Φτfor a.e. τ,
pinning = λ t h 1 = λ t h 1 ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1}=λth1 and, in the slow sector, G , Λ , G , Λ , G,Lambda,dotsG, \Lambda, \ldotsG,Λ, as multipliers associated to their respective allowances.
Outcome. The time-parametrized ascent principle, together with the KKT alignment of the predictive Riesz representer and the throughput derivation, yields A ˙ = i 1 [ H , A ] A ˙ = i 1 [ H , A ] A^(˙)=iℏ^(-1)[H,A]\dot{A} = i \hbar^{-1} [H, A]A˙=i1[H,A] with = λ t h 1 = λ t h 1 ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1}=λth1, and the unbounded-operator hygiene ensures this extends from blocks to the inductive limit.

Appendix F — Technical Lemmas & Quantitative Gaps (W1; microcausality)

F.1 Order-flip ⇒ W 1 W 1 W_(1)W_1W1 gap

For a flip tube with mass parameter θ θ theta\thetaθ and gradient bound L L LLL, define a 1-Lipschitz test via a clipped arrival-time difference. Kantorovich–Rubinstein duality gives W 1 ( P g , P g ~ ) θ Δ v L p o k e W 1 ( P g , P g ~ ) θ Δ v L p o k e W_(1)(P_(g),P_( tilde(g))) >= theta(Deltav_(**))/(L)ℓ_(poke)W_1(P_g, P_{\tilde{g}}) \ \ge \ \theta \, \frac{\Delta v_*}{L} \, \ell_{\rm poke}W1(Pg,Pg~)  θΔvLpoke.

F.2 Microcausality guards

Cone-limited Lieb–Robinson-type bounds for GKSL commutators; principal-symbol lemma enforcing single-cone hyperbolicity.

Appendix G — Horizon Pointer Mechanics: Area Law Details

Block decomposition near horizon; additivity and boundary terms; minimal tilings and constants. Links to Ch. 7 area-law statement.

Appendix H — Hawking Flux Suppression & Microcausality (Asymptotics)

Transfer-envelope asymptotics; LR guards; flux comparison (strict inequality unless λ l e a k = 0 λ l e a k = 0 lambda_(leak)=0\lambda_{\rm leak} = 0λleak=0).

Appendix I — Notation & Functional-Analysis Lemmas

Topologies; Riesz on blocks; l.s.c.; Γ-convergence; W 1 W 1 W_(1)W_1W1 duality; normalized HS inner product; cb-norm basics.

Appendix J — Estimation & Data Kits (Operational Measurability)

J.1 Budget estimation

  • Throughput: calibrate = λ t h 1 = λ t h 1 ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1}=λth1 via two-level Fubini–Study speed.
  • Leakage: estimate j ω ( L j W L j ) j ω ( L j W L j ) sum _(j)omega(L_(j)^(†)WL_(j))\sum_j \omega(L_j^\dagger W L_j)jω(LjWLj) by tomography in the pointer basis; use s.n.f. weight normalization.
  • Complexity: bound B c x B c x B_(cx)\mathcal{B}_{\rm cx}Bcx via local CP decompositions with measured cb-norm surrogates.

J.2 Coherence functional

Design poke ensembles covering a basis of the closure hull; use diamond-norm continuity to control errors. Publish kernels, fits, and code.

J.3 Stop rules and leakage hygiene

Define explicit stop conditions based on primary KPI plateaus; cap leakage channels per ethics/privacy constraints.
Outcome. Ready-to-run templates for experiments that feed the selection program with auditable inputs.

Appendix K — Prediction Worksheets & Fitting Protocols

GW phasing residual; horizon flux suppression; interferometer pointer selection; publishing kit and calibration worksheets aligned with Ch. 10.

Appendix L


L.1 Γ-limit promotion of budget tiles on FRW

Finite-window proofs (LR quasi-factorization, pinching contraction, Dirichlet linearity) extend to FRW via exhaustion by comoving boxes and cone-preserving gauge. First variations commute with the ε 0 ε 0 epsi rarr0\varepsilon \to 0ε0 limit by the localized Mosco/Attouch framework used in Ch. 5. In particular:
  • Locality → global envelope. Tilewise GKSL estimates with leakage/throughput/complexity budgets remain stable under FRW coarse-graining; their per-tile constants are uniform on bounded curvature charts.
  • Continuity of multipliers. Calibration multipliers ( α t h , α c x , α l e a k α t h , α c x , α l e a k alpha_(th),alpha_(cx),alpha_(leak)\alpha_{\rm th}, \alpha_{\rm cx}, \alpha_{\rm leak}αth,αcx,αleak) obtained on finite windows converge along the exhaustion; the slow-sector Γ-limit preserves the Euler–Lagrange form with T e f f T e f f T^(eff)T^{\rm eff}Teff defined by the limit of the fast KKT tensors.

L.2 Planck window constants and the area coefficient

The constant C C CCC in the Planck-window mutual-information bound is fixed by:
  1. Tile spectral gap / log-Sobolev constants (pointer-aligned GKSL),
  2. Lieb–Robinson velocity and mixing length for the fast sector, and
  3. Normalized HS metric calibration that identifies = λ t h 1 = λ t h 1 ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1}=λth1.
Under these inputs, the coefficient of the area law is unique and stable under norm-equivalent budget representatives; replacing any quadratic by a norm-equivalent surrogate leaves the coefficient invariant. Cross-boundary predictive content in the Planck window is area-limited independently of bare UV counts; the leakage budget acts as a soft UV regulator.

L.3 Coherence smoothing vs. inflation (compatibility note)

The “coherence-driven smoothing” mechanism in Ch. 10.3 relies only on LR microcausality and Dirichlet linearity and does not require an inflaton. If an inflaton sector is present in T v i s T v i s T^(vis)T^{\rm vis}Tvis, pointer-aligned leakage contributes an effective dissipative term (friction shift Δ Γ α l e a k Δ Γ α l e a k Delta Gamma propalpha_(leak)\Delta \Gamma \propto \alpha_{\rm leak}ΔΓαleak) while derivation-pricing enforces a kinetic penalty α t h α t h propalpha_(th)\propto \alpha_{\rm th}αth. This yields a controlled channel to warm-inflation-like dynamics without violating LR locality.

L.4 Placement in the manuscript

Insert Appendix L after Appendix K — Prediction Worksheets & Fitting Protocols and before Appendix A1 — Concrete Coherence Functionals, preserving the alphabetical order of lettered appendices while keeping A1 as a special technical appendix.

L.5 Refuter ledger (cosmology & dark sector)

  1. Observation of super-cone signaling or long-range poke effects in early-time cosmological data (violates LR microcausality).
  2. Empirical need for a fourth independent quadratic budget that cannot be represented as a norm-equivalent quadratic under the same symmetries/calibration.
  3. Laboratory observation of basis-invariant decoherence contradicting pointer alignment.
  4. Cosmological data requiring a time-varying \hbar or multipliers inconsistent with Γ-calibration.

Appendix A1 — Concrete Coherence Functionals: Properties & Robustness

We prove l.s.c., concavity under mixing, and risk-sensitive convergence for the two concrete CLs of §1.2, and then show robustness within a broader admissible family.

A1.1 Setup and notation

Let P P P\mathscr PP be the admissible poke cone (closed under composition/mixing), and let T T T\mathscr TT be a finite family of protocols (finite observables and post-processings). A mixed pattern is a probability measure μ μ mu\muμ on a finite set of base patterns { A j } { A j } {A_(j)}\{A_j\}{Aj}; its dynamics are mixtures of the base dynamics.

A1.2 CA toy: CL C A CL C A CL_(CA)\mathrm{CL}_{\rm CA}CLCA is l.s.c. and concave (under mixing)

Lemma A1.1 (l.s.c.). On a fixed n , T n , T n,Tn,Tn,T, the maps ( A , Φ ) S A , Φ T C A , M A , Φ T C A , L A , Φ T C A ( A , Φ ) S A , Φ T C A , M A , Φ T C A , L A , Φ T C A (A,Phi)|->S_(A,Phi)^(T_(CA)),M_(A,Phi)^(T_(CA)),L_(A,Phi)^(T_(CA))(A,\Phi)\mapsto S^{T_{\rm CA}}_{A,\Phi},\,M^{T_{\rm CA}}_{A,\Phi},\,L^{T_{\rm CA}}_{A,\Phi}(A,Φ)SA,ΦTCA,MA,ΦTCA,LA,ΦTCA are continuous in the product of discrete/trace topologies; hence any finite affine combination is continuous, and CL C A = max T C A , u F T C A , u CL C A = max T C A , u F T C A , u CL_(CA)=max_(T_(CA),u)F_(T_(CA),u)\mathrm{CL}_{\rm CA}=\max_{T_{\rm CA},u}F_{T_{\rm CA},u}CLCA=maxTCA,uFTCA,u is l.s.c.
Proof. Each is a cylinder-event probability or bounded expectation on a finite Markov chain parameterized by finitely many transition probabilities; continuity follows from finite-dimensionality. Max over a finite set preserves l.s.c. ∎
Lemma A1.1′ (concavity under mixing). If μ μ mu\muμ is a distribution over base patterns { A j } { A j } {A_(j)}\{A_j\}{Aj}, then A E μ [ F T C A , u ( A , Φ ) ] A E μ [ F T C A , u ( A , Φ ) ] A|->E_(mu)[F_(T_(CA),u)(A,Phi)]A\mapsto \mathbb E_\mu[F_{T_{\rm CA},u}(A,\Phi)]AEμ[FTCA,u(A,Φ)] is affine in μ μ mu\muμ; the pointwise maximum over ( T C A , u ) ( T C A , u ) (T_(CA),u)(T_{\rm CA},u)(TCA,u) of affine maps is concave. Hence A CL C A ( A , Φ ) A CL C A ( A , Φ ) A|->CL_(CA)(A,Phi)A\mapsto \mathrm{CL}_{\rm CA}(A,\Phi)ACLCA(A,Φ) is concave in μ μ mu\muμ.
Proof. Linearity in μ μ mu\muμ is immediate (law of total expectation). Pointwise max of affine maps is concave. ∎

A1.3 Quantum toy: CL Q CL Q CL_(Q)\mathrm{CL}_{\rm Q}CLQ is l.s.c. and concave (under mixing)

Lemma A1.2 (l.s.c.). The map N N ( Δ ) 1 N N ( Δ ) 1 N|->||N(Delta)||_(1)\mathcal N\mapsto \|\mathcal N(\Delta)\|_1NN(Δ)1 is continuous in the diamond norm; taking inf Φ P inf Φ P ¯ i n f_(Phi in bar(P))\inf_{\Phi\in\overline{\mathscr P}}infΦP yields an l.s.c. functional CL Q CL Q CL_(Q)\mathrm{CL}_{\rm Q}CLQ on ( A , Φ ) ( A , Φ ) (A,Phi)(A,\Phi)(A,Φ).
Proof. For fixed Δ Δ Delta\DeltaΔ, ( Δ ) 1 Δ 1 ( Δ ) 1 Δ 1 ||*(Delta)||_(1) <= ||*||_(diamond)||Delta||_(1)\|\cdot(\Delta)\|_1\le \|\cdot\|_\diamond\|\Delta\|_1(Δ)1Δ1. Continuity in ||*||_(diamond)\|\cdot\|_\diamond and the infimum over a compact poke class give l.s.c. ∎
Lemma A1.2′ (concavity under mixing). If A μ A μ A∼muA\sim \muAμ is a mixture of base channels { N A j , Φ } { N A j , Φ } {N_(A_(j),Phi)}\{\mathcal N_{A_j,\Phi}\}{NAj,Φ}, then E μ [ N A , Φ ( Δ ) ] 1 E μ [ N A , Φ ( Δ ) 1 ] E μ [ N A , Φ ( Δ ) ] 1 E μ [ N A , Φ ( Δ ) 1 ] ||E_(mu)[N_(A,Phi)(Delta)]||_(1) <= E_(mu)[||N_(A,Phi)(Delta)||_(1)]\|\mathbb E_\mu[\mathcal N_{A,\Phi}(\Delta)]\|_1\le \mathbb E_\mu[\|\mathcal N_{A,\Phi}(\Delta)\|_1]Eμ[NA,Φ(Δ)]1Eμ[NA,Φ(Δ)1]. Thus A F Q ( A , Φ ) A F Q ( A , Φ ) A|->F_(Q)(A,Phi)A\mapsto F_{\rm Q}(A,\Phi)AFQ(A,Φ) is concave in μ μ mu\muμ, and so is A CL Q ( A , Φ ) A CL Q ( A , Φ ) A|->CL_(Q)(A,Phi)A\mapsto \mathrm{CL}_{\rm Q}(A,\Phi)ACLQ(A,Φ) after inf Φ inf Φ i n f _(Phi)\inf_\PhiinfΦ.
Proof. Triangle inequality and Jensen for the norm; the affine dependence on μ μ mu\muμ gives concavity. ∎

A1.4 Risk-sensitive worst-case limit

Proposition A1.2 (risk-sensitive limit). For either concrete CL and any poke law Π Π Pi\PiΠ, define CL β ( A ) := β 1 log E Φ Π exp ( β CL ( A , Φ ) ) CL β ( A ) := β 1 log E Φ Π exp ( β CL ( A , Φ ) ) CL_(beta)(A):=beta^(-1)log E_(Phi∼Pi)exp(betaCL(A,Phi))\mathrm{CL}_\beta(A):=\beta^{-1}\log\mathbb E_{\Phi\sim\Pi}\exp(\beta\,\mathrm{CL}(A,\Phi))CLβ(A):=β1logEΦΠexp(βCL(A,Φ)). Then CL β inf Φ P CL ( A , Φ ) CL β inf Φ P ¯ CL ( A , Φ ) CL_(beta)↘i n f_(Phi in bar(P))CL(A,Phi)\mathrm{CL}_\beta\searrow \inf_{\Phi\in\overline{\mathscr P}}\mathrm{CL}(A,\Phi)CLβinfΦPCL(A,Φ) pointwise and epi-converges as β β beta rarr oo\beta\to\inftyβ.
Proof. Standard log-moment generating function convergence (Varadhan/Cramér type) on bounded functionals; epi-convergence follows from monotone convergence properties. ∎

A1.5 Robustness: no fine-tuning

Proposition A1.3 (equivalence class of CLs). Let F F F\mathcal FF be the family
[
\mathcal F:=\Big{\mathrm{CL}s(A,\Phi):=\sup{T\in\mathscr T}\mathbb E\big[s_T(Z_{A,\Phi})\big]\ \Big|\ s_T:\mathcal Z_T\to\mathbb R\ bounded, concave, strictly increasing in success features\Big}.
]
On any bounded window and admissible poke cone, there exist constants 0 < c 1 c 2 < 0 < c 1 c 2 < 0 < c_(1) <= c_(2) < oo0<c_1\le c_2<\infty0<c1c2< and increasing functions h 1 , h 2 h 1 , h 2 h_(1),h_(2)h_1,h_2h1,h2 such that for any s , s s , s s,s^(')s,s's,s in the family,
h 1 ( CL s ( A , Φ ) ) CL s ( A , Φ ) h 2 ( CL s ( A , Φ ) ) , h 1 ( CL s ( A , Φ ) ) CL s ( A , Φ ) h 2 ( CL s ( A , Φ ) ) , h_(1)(CL_(s)(A,Phi)) <= CL_(s^('))(A,Phi) <= h_(2)(CL_(s)(A,Phi)),h_1\!\big(\mathrm{CL}_s(A,\Phi)\big)\ \le\ \mathrm{CL}_{s'}(A,\Phi)\ \le\ h_2\!\big(\mathrm{CL}_s(A,\Phi)\big),h1(CLs(A,Φ))  CLs(A,Φ)  h2(CLs(A,Φ)),
uniformly on compact sets. Consequently, maximizing CL s λ i B i CL s λ i B i CL_(s)-sumlambda _(i)B_(i)\mathrm{CL}_s-\sum\lambda_i B_iCLsλiBi or CL s λ i B i CL s λ i B i CL_(s^('))-sumlambda _(i)B_(i)\mathrm{CL}_{s'}-\sum\lambda_i B_iCLsλiBi has the same maximizers in the Γ Γ Gamma\GammaΓ-limit, and the KKT multipliers (e.g. = λ t h 1 = λ t h 1 ℏ=lambda_(th)^(-1)\hbar=\lambda_{\rm th}^{-1}=λth1) agree.
Proof sketch. Bounded concave scores on finite observables are bi-Lipschitz equivalent up to increasing transforms on compact ranges; the sup over a finite T T T\mathscr TT preserves equivalence. Stability of maximizers and multipliers follows from Γ Γ Gamma\GammaΓ-equivalence and envelope regularity established in Ch. 4–5. ∎
Corollary A1.4 (pointer robustness). For s s sss chosen as the quantum reliability score F Q F Q F_(Q)F_{\rm Q}FQ or any strictly increasing concave transform thereof, leakage-budget minimizers align with the W W WWW-eigenbasis (pointer basis), independent of the particular s s sss.